[Shoshichi Kobayashi] Transformation Groups in Dif(BookFi.org)

Shoshichi Kobayashi

Transformation Groups in Differential Geometry

Reprint of the 1972 Edition

Springer

Shoshichi Kobayashi Department of Mathematics, University of California Berkeley, CA 94720-3840 USA

Originally published as Vol. 70 of the Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd sequence

Mathematics Subject Classification (1991): Primary 53C20, 53C10, 53C55, 32M05, 32)15, 57S15 Secondary 53C15, 53A10, 53A20, 53A30, 32H20, 58D05

ISBN 3-540-58659-8 Springer-Verlag Berlin Heidelberg New York

Photograph by kind permission of George Bergman

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SPIN 10485278 41/3140 - 5 4 3 2 1 0 - Printed on acid-free paper

Shoshichi Kobayashi

Transformation Groups in Differential Geometry

Springer-Verlag Berlin Heidelberg New York 1972

Shoshichi Kobayashi University of California, Berkeley, California

AMS Subject Classifications (1970):

Primary 53 C 20, 53 C 10, 53 C 55, 32 M 05, 32 J 25, 57 E 15

Secondary 53 C 15, 53 A 10, 53 A 20, 53 A 30, 32 H 20, 58 D 05

ISBN 3-540-05848-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-05848-6 Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright All rights are reserved, whether the whole or part of the material Is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, repro-duction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 0) by Springer-Verlag Berlin Heidel-berg 1972. Library of Congress Catalog Card Number 72-80361. Printed in Germany. Printing and bin-ding: Universitatsdruckerei H. Startz AG, Wfirzburg

Preface

Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc-tures. All geometric structures are not created equal; some are creations of gods while others are products of lesser human minds. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures.

Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure. Basic theorems in this regard are presented in §§ 3, 4 and 5. The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo-metric structures in a unified manner. In § 8, we sketch the relationship between the two concepts. Chapter I is so arranged that the reader who is primarily interested in Riemannian, complex, conformal and projective structures can skip §§ 5, 6, 7 and 8. This chapter is partly based on lec-tures I gave in Tokyo and Berkeley in 1965.

Contents of Chapters II and III should be fairly clear from the section headings. It should be pointed out that the results in §§ 3 and 4 of Chapter II will not be used elsewhere in this book and those of §§ 5 and 6 of Chapter II will be needed only in §§ 10 and 12 of Chapter III. I lectured on Chapter II in Berkeley in 1968; Chapter II is a faithful version of the actual lectures.

Chapter IV is concerned with automorphisms of affine, projective and conformal connections. We treat both the projective and the con-formal cases in a unified manner.

Throughout the book, we use Foundations of Differential Geometry as our standard reference. Some of the referential results which cannot be found there are assembled in Appendices for the convenience of the reader.

As its 'title indicates, this book is concerned with the differential geometric aspect rather than the differential topological or homological

VI Preface

aspect of the theory of transformation groups. We have confined our-selves to presenting only basic results, avoiding difficult theorems. To compensate for the omission of many interesting but difficult results, we have supplied the reader with an extensive list of references.

We have not touched upon homogeneous spaces, partly because they form an independent discipline of their own. While we are interested in automorphisms of given geometric structures, the differential geometry of homogeneous spaces is primarily concerned with geometric objects which are invariant under given transitive transformation groups. For the convenience of the reader, the Bibliography includes papers on the geometry of homogeneous spaces which are related to the topics discussed here.

In concluding this preface, I would like to express my appreciation to a number of mathematicians: Professors Yano and Lichnerowicz, who interested me in this subject through their lectures, books and papers; Professor. Ehresmann, who taught me jets, prolongations and infinite pseudo-groups; K. Nomizu, T. Nagano and T. Ochiai, my friends and collaborators in many papers; Professor Matsushima, whose recent monograph on holomorphic vector fields influenced greatly the presen-tation of Chapter III; Professor Howard, who kindly made his manu-script on holomorphic vector fields available to me. I would like to thank Professor Remmert and Dr. Peters for inviting me to write this book and for their patience.

I am grateful also to the National Science Foundation for its un-failing support given to me during the preparation of this book.

January, 1972 S. Kobayashi

Contents

I. Automorphisms of G-Structures 1

1. G-Structures 1 2. Examples of G-Structures 5 3. Two Theorems on Differentiable Transformation Groups. • 13 4. Automorphisms of Compact Elliptic Structures 16 5. Prolongations of G-Structures 19 6. Volume Elements and Symplectic Structures 23 7. Contact Structures 28 8. Pseudogroup Structures, G-Structures and Filtered Lie Alge-

bras 33

II. Isometries of Riemannian Manifolds 39

1. The Group of Isometries of a Riemannian Manifold. . . 39 2. Infinitesimal Isometries and Infinitesimal Affine Trans-

formations 42 3. Riemannian Manifolds with Large Group of Isometries . 46 4. Riemannian Manifolds with Little Isometries 55 5. Fixed Points of Isometries 59 6. Infinitesimal Isometries and Characteristic Numbers . . • 67

III. Automorphisms of Complex Manifolds 77

1. The Group of Automorphisms of a Complex Manifold . . 77 2. Compact Complex Manifolds with Finite Automorphism

Groups 82 3. Holomorphic Vector Fields and Holomorphic 1-Forms . • 90 4. Holomorphic Vector Fields on Kahler Manifolds . . . • 92 5. Compact Einstein-Kahler Manifolds 95 6. Compact Kahler Manifolds with Constant Scalar Curvature 97 7. Conformal Changes of the Laplacian 100 8. Compact Kahler Manifolds with Nonpositive First Chern

Class 103

VIII Contents

9. Projectively Induced Holomorphic Transformations. . . . 106 10. Zeros of Infinitesimal Isometries 112 11. Zeros of Holomorphic Vector Fields 115 12. Holomorphic Vector Fields and Characteristic Numbers. 119

IV. Affine, Conformal and Projective Transformations 122

1. The Group of Affine Transformations of an Affinely Con- nected Manifold 122

2. Affine Transformations of Riemannian Manifolds 125 3. Cartan Connections 127 4. Projective and Conformal Connections 131 5. Frames of Second Order 139 6. Projective and Conformal Structures 141 7. Projective and Conformal Equivalences 145

Appendices 150

1. Reductions of 1-Forms and Closed 2-Forms 150 2. Some Integral Formulas 154 3. Laplacians in Local Coordinates 157 4. A Remark on d'd"-Cohomology 159

Bibliography 160

Index 181

I. Automorphisms of G-Structures

1. G-Structures

Let M be a differentiable manifold of dimension n and L(M) the bundle of linear frames over M. Then L(M) is a principal fibre bundle over M with group GL(n; R). Let G be a Lie subgroup of GL(n; R). By a G-struc-ture on M we shall mean a differentiable subbundle P of L(M) with structure group G.

There are very few general theorems on G-structures. But we can ask a number of interesting questions on G-structures, and they are often very difficult even for some specific G. It is therefore essential for the study of G-structures to have familiarity with a number of examples.

In general, when M and G are given, there may or may not exist a G-structure on M. If G is a closed subgroup of GL(n; R), the existence problem becomes the problem of finding cross sections in a certain bundle. Since GL(n; R) acts on L(M) on the right, a subgroup G also acts on L(M). If G is a closed subgroup of GL(n; R), then the quotient space L(M)/G is the bundle with fibre GL(n; R)/G associated with the principal bundle L(M). It is then classical that the G-structures on M are in a natural one-to-one correspondence with the cross sections

M L(M)/G

(see, for example, Kobayashi-Nomizu [1, vol. 1; pp. 57-58]). The so-called obstruction theory gives necessary algebraic-topological condi-tions on M for the existence of a G-structure (see, for example, Steen-rod [1]).

A G-structure P on M is said to be integrable if every point of M has a coordinate neighborhood U with local coordinate system x', , xn such that the cross section (a/3x1 , Olaf) of L(M) over U is a cross section of P over U. We shall call such a local coordinate system x1 , , xn admissible with respect to the given G-structure P. If x', , x" and y', , y" are two admissible local coordinate system in open sets U and 1/ respectively, then the Jacobian matrix (a yi/axi)i , is in G at each point of U n V

2 L Automorphisms of G-Structures

Proposition 1.1. Let K be a tensor over the vector space R" (j. e., an element of the tensor algebra over R") and G the group of linear transfor-mations 01W leaving K invariant. Let P be a G-structure on M and K the tensor field on M defined by K and P in a natural manner (see the proof below). Then P is integrable if and only if each point of M has a coordinate neighborhood with local coordinate system x', x" with respect to which the components of K are constant functions on U.

Proof We give the definition of K although it is more or less obvious. At each point x of M, we choose a frame u belonging to P. Since u is a linear isomorphism of R" onto the tangent space T(M), it induces an isomorphism of the tensor algebra over Rn onto the tensor algebra over T(M). Then Kx is the image of K under this isomorphism. The invariance of K by G implies that K x is defined independent of the choice of u.

Assume that P is integrable and let x', , f be an admissible local coordinate system. From the construction above, it is clear that the components of K with respect to x', , f coincide with the components of K with respect to the natural basis in R" and, hence, are constant functions.

Conversely, let x1 , , f be a local coordinate system with respect to which K has constant components. In general, this coordinate system is not admissible. Consider the frame (/ax', a/axn) at the origin of this coordinate system. By a linear change of this coordinate system, we obtain a new coordinate system y', , y" such that the frame (ô/ay', ..., amyl) at the origin belongs to P. Then K has constant components with respect to y', , y". These constant components coincide with the com-ponents of K with respect to the natural basis of R" since Way', abayn) at the origin belong to P. Let u be a frame at x e U belonging to P. Since the components of K with respect to u coincide with the components of K with respect to the natural basis of R" and, hence, with the compo-nents of K with respect to (a/ay', ...,alayn), it follows that the frame (ô14', ...,a/ayn) at x coincides with u modulo G and, hence, belongs to P. q e. d.

Proposition 1.2. If a G-structure P on M is integrable, then P admits a torsionfree connection.

Proof Let U be a coordinate neighborhood with admissible local coordinate system x', , xn. Let cou be the connection form on Pi U defining a fiat affine connection on U such that alax, ,...,alaxn are parallel. We cover M by a locally finite family of such open sets U. Taking a partition of unity tfu} subordinate to { U), we define a desired

1. G-Structures 3

connection form co by

a)=En*fu • wu,

where 7t: P M is the projection. q.e.d.

In some cases, the converse of Proposition 1.2 is true. For such examples, see the next section.

Let P and P' be G-structures over M and M'. Letfbe a diffeomorphism of M onto M' and f* : L(M)--* L(M) the induced isomorphism on the bundles of linear frames. If f, maps P into P', we call f an isomorphism of the G-structure P onto the G-structure P. If M=M' and P = P', then an isomorphism f is called an automorphism of the G-structure P.

A vector field X on M is called an infinitesimal automorphism of a G-structure P if it generates a local 1-parameter group of automorphisms of P.

As in Proposition 1.1, we consider those G-structuresdefined by a tensor K. Then the following proposition is evident.

Proposition 1.3. Let K be a tensor over the vector space Rn and G the group of linear transformations of Rn leaving K invariant. Let P be a G-structure on M and K the tensor field on M defined by K and P. Then

(1) A diffeomorphismf: M M is an automorphism of P if and only iff leaves K invariant;

(2) A vector field X on M is an infinitesimal automorphism of P if and only if Lx K =0, where Lx denotes the Lie derivation with respect to X.

We shall now study the local behavior of an infinitesimal automor-phism of an integrable G-structure. Without loss of generality, we may assume that M =Rn with natural coordinate system x 1, , xn and P =Rn x G. Let X be a vector field in (a neighborhood of the origin of) Rn and expand its components in power series:

c° 1 E E x31 xik,

kr. 0 — • )1, ..., fic.1

where 4 are symmetric in the subscripts j1 , Since X is an infinitesimal automorphism of P if and only if the matrix (a viaxi) belongs to the Lie algebra g of G, we may conclude that X is an infini-tesimal automorphism of P if and only if, for each fixed j 2 , ...,jk , the matrix (4 n belongs to the Lie algebra g. This motivates the following definition.

4 1. Automorphisms of G-Structures

Let g be a Lie subalgebra of gl(n; R). For k =0, 1, 2, ..., let gk be the space of symmetric multilinear mappings

t: R'' x - - • x R" R" ,—....,—.

(k +1)-times

such that, for each fixed v 1 , ... , vic e R", the linear transformation

ye R" -- t(v, v1 , ..., vk)eR"

belongs to g. In particular, go = g. We call gk the k-th prolongation of g. The first integer k such that gk =0 is called the order of g. If gk =0, then 9k+1=gk+ 2 = • • • =0. If g,,0 for all k, then g is said to be of infinite type.

Proposition 1.4. A Lie algebra g c gl (n; R) is of infinite type if it contains a matrix of rank 1 as an element.

Proof Let e be a nonzero element of R" and a a nonzero element of the dual space of W. Then the linear transformation defined by

veR"-- <a, v> eeR"

is of rank 1, and conversely, every linear transformation of rank 1 is given as above. Assume that the transformation above belongs to g. For each positive integer k, we define

t(vo , V 1 , ... , vk) = <a, vo > <a, v 1 > - - - <a, vk > e, vi e R".

Then t is a nonzero element of gk . q.e. d.

We say that a Lie algebra g c gl(n; R) is elliptic if it contains no matrix of rank 1. Proposition 1.4 means that if g is of finite order, then it is elliptic.

Each Lie subalgebra g of gl(n; R) gives rise to a graded Lie algebra CO

E 9k, where g_ 1 =Rn. The bracket of te gp and t'Egq is defined by k = —1

1 E vj ... , vj), .. , [t, tl (vo, vi, ... , vp+ q)=

p! (q + I)! t(e( o , vj,,, .vi„ +„)

1 E (p+ I)! q!

c0 , ... , vkp ), v k,„ ... , vkp+ ,7)•

In particular, if tEgp , p. 0, and v e g_ i =R", then

[t, v](v i , ... , vp)=t(v, v 1 , ... , v a).

We explicitly set [g_ 1 , g_ 1 ] =0. This definition is motivated by the following geometrical consideration. Suppose t=(4 ... jp )e gp and t' = (b iko... kci )e gq in terms of components and consider the corresponding

2. Examples of G-Structures 5

vector fields: 1 x. E d xi

a ° xi

(p+1)! k..-JP *** P axi)

1 a Y (q±1)!

co Then [X, Y] corresponds to [t, Thus, the graded Lie algebra E gk may be considered as the Lie algebra of infinitesimal automorphisms

. a E — with polynomial components c of the flat G-structure P =

axl R" x G on Rn.

For a survey on G-structures, see expository articles of Chern [1], [2]; the latter contains an extensive list of publications on the subject. See also Sternberg's book [1], A. Fujimoto [2], [3], Bernard [1].

The group of automorphisms of a compact elliptic structure or a G-structure of fmite type will be shown to be a Lie transformation group (see §§ 4 and 5, respectively). These two cases cover a substantial number of interesting geometric structures whose automorphism groups are Lie groups. By considering G-structures of higher degree, we can bring such structures as projective structures under this general scheme (see § 8 of this chapter and Chapter IV). The group of automorphisms of a bounded domain or a similar complex manifold is also a Lie group (see § 1 of Chapter III), but this does not come under the general scheme. This book does not touch area-measure structures (Brickell [1]), nor pseudo-con-formal structures of real hypersurfaces in complex manifolds (Morimoto-Nagano [1], Tanaka [3]) although automorphism groups of these struc-tures are usually Lie groups.

2. Examples of G-Structures

Example 2.1. G= GL(n; R) and g = gl (n; R). The Lie algebra g contains a matrix of rank 1 and is of infinite type. A G-structure on M is nothing but the bundle L(M) of linear frames and is obviously integrable. Every diffeomorphism of M onto itself is an automorphism of this G-structure and every vector field on M is an infinitesimal automorphism.

Example 2.2. G = GL + (n; R) and g = g! (n; R), where GL + (n; R) means the group of matrices with positive determinant. The Lie algebra g is of infinite type. A manifold M admits a GL + (n; R)-structure if and only if it is orientable; this is more or less the definition of orientability. A GL + (n; R)-structure on M may be considered as an orientation of M and is obviously integrable. A diffeomorphism of M onto itself is an

6 I. Automorphisms of G-Structures

automorphism of a GL + (n; R)-structure if and only if it is orientation preserving. Every vector field on M is an automorphism since every one-parameter group of transformations is orientation preserving.

Example 2.3. G= SL(n; R) and g _—%1(n; R). Again, g contains a matrix of rank 1 and is of infinite type. The natural action of GL(n; R) on R" induces an action of GL(n; R) on A" Rn such that

A v det (A) • v for A e GL(n ; R) and v ez1" Rn.

The group GL(n; R) is transitive on A" R"— {0} with isotropy subgroup SL(n; R) so that A" — {0} = GL(n; R)/SL(n; R). It follows that the cross sections of the bundle L(M)/SL(n; R) are in one-to-one correspondence with the volume elements of M, L e., the n-forms on M which vanish nowhere. In other words, an SL(n; R)-structure is nothing but a volume element on M. It is clear that M admits an SL(n; R)-structure if and only if it is orientable. We claim that every SL(n; R)-structure is integrable. Indeed, let U be a coordinate neighborhood with local coordinate system

x" and let (p =f dx 1 A • • • A dxn be the volume element correspond-ing to the given SL(n; R)-structure. Let = y 1 (x', , xn) be a function such that ay 1lax1 =f Then

= f dx l dx" = dy l A dX 2 A • • • A dx",

which shows that the coordinate system y 1 , x2 , x" is admissible with respect to the given SL(n; R)-structure. A diffeomorphism of M onto itself is an automorphism of the SL(n; R)-structure if and only if it preserves the volume element (p. Let X be a vector field on M. The function (

5X defined by

Lx = (6 X) • (p

is called the divergence of X with respect to (p. Clearly, X is an infini-tesimal automorphism of the SL(n; R)-structure if and only if (5 X =0. For SL(n; R)-structures, see § 6.

Example 2.4. G = GL(m; C) and g = gI(m; C). We consider GL(m; C) (resp. gl (m; C)) as a subgroup of GL(2 m; R) (resp. a subalgebra of gl(2 m; R)) in a natural manner, i.e.,

A A l +i A 2 EGL(m; C) _ ( ' A2 )

_ A2 Ai eGL(2m, R)

or gI(m; C) or gl (2 m ; R) .

Let z1 , , zm be the natural coordinate system in Cm and zi= xi+ i xrn+ j =1, , m. Then the identification e" =R 2 ' given by

z 1, ... es) ..., X

2 m

)


induces the preceding injections

GL(m; C) --*GL(2 m; R) and gl(m; gl(2m; R).

The multiplication by i in Cm, i.e.,

, zm)--* izm),

induces a linear transformation

xm, xm+ 1 , ..., X 2 m) .--*( -Xm+1 , - X2 m, X 1 , xm)

of R2 m, which will be denoted by J. Since i 2 = —1, we have J 2 = —I. In matrix form,

J = (O — I

I 01

The group GL(m; C) (resp. the algebra gi(m; C)), considered as a sub-group of GL(2 m; R) (resp. a subalgebra of gl (2 m; R)), is given by

GL(m, C)= {AEGL(2m; R); AJ=J A}

gl(m; C). {Aeg1(2m; R); AJ =JA}.

Since gk consists of all symmetric multilinear mappings of Cm x • • • x Cm (k + 1 times) into Cm, the Lie algebra g is of infinite type. Every element of g, considered as an element of gl(2 m; R) is of even rank. Hence, g is elliptic. The GL(m; C)-structure on a manifold M (of dimension 2m) are in one-to-one correspondence with the tensor field J of type (1, 1) on M such that

Jx Jx = — Ix (or simply, J J = — I),

where ./x is the endomorphism of the tangent space TX(M) given by J and Ix is the identity transformation of T(M). The correspondence is given as follows. Given a tensor field J with J 0J. — I, we consider, at each point x of M, only those linear frames u: R 2 m Tx (M) satisfying u 0 J = Jo The subbundle of L(M) thus obtained is the corresponding GL(m; 0-structure on M. A tensor field J with Jo J = — I or the cor-responding GL(m; C)-structure is called an almost complex structure. We claim that an almost complex structure is integrable (as a GL(m; C)- structure) if and only if it comes from a complex structure. (Before we explain this statement, we should perhaps remark an almost complex structure J is often called integrable if a certain tensor field of type (1, 2), called the torsion or Nijenhuis tensor, vanishes.) It is a deep result of Newlander and Nirenberg [1] that the two definitions coincide. For the real analytic case, see, for instance, Kobayashi-Nomizu [1, vol. 2; p. 321]. The theorem of Newlander-Nirenberg is equivalent to the statement that an GL(m; C)- structure is integrable if and only if it admits a torsionfree affine connec-


tion (see Fr6hlicher [1]). Let M be a complex manifold of complex dimension m with local coordinate system z 1, z'n where zi= xi+ i yi. We have the natural almost complex structure J on M defined by

J(a/axi)= 3/3y1 j= 1, , m,

J (a/a yi)= —a/axi j= 1, , m.

The almost complex structure J thus obtained is integrable since

(a/ax', ,

gives a local cross section of the GL(m, C)-structure defined by J. Con-versely, if an almost complex structure J is integrable as a GL(m; C)- structure and if x1 , , x2 m is an admissible local coordinate system, then J(3/3x-1)=alaxm±i and J(a/axm+i)= —a/axi for j= 1, ..., m. If we set zi = xi + i xm+i, then the complex coordinate system z1 , , zm turns M into a complex manifold. A diffeomorphismf of M onto itself is an auto-morphism of an almost complex structure J if and only if f* 0 J=J of* , where f* : T(M)--* T(M) is the differential off If J is integrable, an auto-morphism f is nothing but a holomorphic diffeomorphism. A vector field X on M is an infinitesimal automorphism of an almost complex structure J if and only if

[X,JY]=J([X, Y]) for all vector field Yon M.

For further properties of an almost complex structure, see Kobayashi-Nomizu [1; Chapter IX].

Example 2.5. G= 0(n) and g = o(n). The Lie algebra g is of order 1. Let te g l and (4 k) the components of t. By definition, t"ik = tip Since o(n) consists of skew-symmetric matrices, we have t.iik = — tf k . Hence,

4k=tf-tjtjt j t j

thus proving ti,, =0. To each Riemannian metric on M, there corresponds the bundle of orthonormal frames over M. This gives a one-to-one corre-spondence between the Riemannian metrics on M and the 0(n)-struc-tures on M. An 0(n)-structure is integrable if and only if the corresponding Riemannian metric is flat, i. e., it has vanishing curvature. An automor-phism of an 0(n)-structure is an isometry of the corresponding Rieman-nian metric. An infinitesimal automorphism of an 0(n)-structure is an infinitesimal isometry or Killing vector field. We shall discuss isometries and Killing vector fields in detail later (see Chapter II).

More generally, let G= 0(p, q), n = p + q, be the orthogonal group

p2 p2 defined by a quadratic form ±u u- Then o (p, q) is

also of order I. There is a natural one-to-one correspondence between


the pseudo-Riemannian metrics of signature q on M and the O(p, q)- structures on M. An 0(p, q)-structure is integrable if and only if the corresponding pseudo-Riemannian metric has vanishing curvature. It should be remarked that, although every paraconipact manifold admits a Riemannian metric, it may not in general admit a pseudo-Riemannian metric of signature q for q #0, n. For automorphism of pseudo-Rieman-nian manifolds, see Tanno. [1, 2].

Example 2.6. G = CO (n) and g = co (n), n 3. By definition,

CO(n)= {ileGL(n; R); = c I, cell, c>0},

co(n)= (Aegl(n; R); +A=ci,

Thus, CO (n) = 0(n) x R+ and co (n) = o (n) + R, where R+ denotes the multiplicative group of positive real numbers. The Lie algebra co (n) is of order 2 and the first prolongation g1 is naturally isomorphic to the dual space IV* of R. To determine g 1 , let t=(t .k) be an element of g 1 . Since the kernel of the homomorphism A e co(n) --*trace(A)eR is pre-cisely o(n) and since o(n) is of order 1, the linear mapping

t=(ti k)e = (-1 E t k) eRn* n

is injective. The kernel is the first prolongation of o (n). (The factor of 1

—n

is, of course, not important). To see that this mapping is also surjective,

we have only to observe that = (ic) is the image of t with tj k = 614+ bik To prove g2 =0, let t=(tilik)eg2 . For each fixed k, tl'ik may be considered as the components of an element in g1 and hence can be uniquely written

tiljk =4311 jk+ (51.1 ik -45 13k•

Since tIlik must be symmetric in all lower indices, we have E = E

h h

from which follows jk = From — nt jk =E ?itch, we obtain (n 2) fic = h h

— 6 ih• E from which follows (n —2) .hh = n hh and, hence, h h h

E hh =0. From (n —2) j k = — 6 Pc' Eh h =0 and n 3, we conclude c J k =0. h h

(The reader who prefers an index-free proof is referred to Kobayashi-Nagano [3, III; p. 686].) A CO (n)-structure is called a conformal structure. We say that two Riemannian metrics on M are conformally equivalent if one is a multiple of the other by a positive function. The conformal equivalence classes of Riemannian metrics on M are in a natural one-to-one correspondence with the CO(n)-structure on M. A conformal


structure is integrable if and only if any Riemannian metric corresponding to the structure is locally conformally equivalent to (dx 1 )2 + • • • +(dx") 2

with respect to a suitable local coordinate system x', , xn. Thus, a conformal structure is integrable if and only if it is conformally flat in the classical sense (see Eisenhart [1]). Consequently, the integrability of a conformal structure is equivalent to the vanishing of the so-called conformal curvature tensor of Weyl (provided n 3). Given a Riemannian metric g on M, a diffeomorphismf of M onto itself (resp. a vector field X on M) is a conformal transformation, i. e., an automorphism of the con-formal structure (resp. an infinitesimal conformal transformation, i. e., an infinitesimal automorphism of the conformal structure) if and only if

f * g=p . g (resp. Lx g = a • g),

where p (resp. a) is a positive function (resp. a function) on M. Conformal structures and their automorphisms will be discussed in Chapter IV.

The reason we excluded the case n =2 is that CO (2) (resp. co (2)) is naturally isomorphic to GL(1; C) (resp. 91(1; C)). For this reason, the conformal differential geometry in dimension 2 differs significantly from that in higher dimensions. In particular, we note that every CO (2)-struc-ture, e., GL(1 ; C)-structure is integrable; this is nothing but the existence of isothermal coordinate systems.

The results for CO (n)-structures can be easily generalized to CO (p, q)- structures, where CO (p, q)= O (p, q) x R+ is defined by a quadratic form of signature q.

Example 2.7. G = U(m) and g = u (m). Since u(m) is a subalgebra of 0(2 m) which is of order 1 (cf. Examples 2.4 and 2.5), it is also of order 1. A U(m)-structure on a 2 m-dimensional manifold M is called an almost hermitian structure; it consists of an almost complex structure and a hermitian metric. Since U(m)= GL(m ; C) 0(2 m), a U(m)-structure may be considered as an intersection of a GL(m ; C)-structure and an 0(2 m)-structure. A U(m)-structure is integrable if and only if the underlying almost complex structure is integrable (so that M is a complex manifold) and the hermitian metric has vanishing torsion and curvature. A diffeo-morphism of M onto itself is an automorphism of a U(m)-structure if and only if it is an automorphism of the underlying GL(m, C)- and 0(2 m)-structures. Similarly, for an infinitesimal automorphism. For automor-phisms of hermitian manifolds, see Tanno [3].

Example 2.& G= Sp (m; R) and g =%p (m; R). We recall that Sp (m; R) is the group of linear transformations of R 2 rn leaving the form

Ul A Um+1 ± • —Fan A 142m


invariant, where u', ... , u'm is the natural coordinate system in 11 2 m. In other words,

Sp (m; R)= {A e GL(2 m; R); `AJ A = J} ,

sp (m; R)= {Aeg1(2m; R); 'AJ +J A=0} ,

J = (0 I — 01) '

Since sp(m; R) consists of matrices of the form

A= A (Ai ) A2 : 4.1„ EA A t A 'WI L1.1 11 2 = 112 and 'A 3 = A3 9

113 ii i

it contains an element of rank 1 and, hence, is of infinite type. The Sp(m, R)-structures on a 2m-dimensional manifold M are in a natural one-to-one correspondence with the 2-forms co on M of maximum rank (j. e., com 40 everywhere).

Since both GL(m; C) and Sp (m; R) contain U(m) as a maximal com-pact subgroup, a manifold M admits an Sp (m; R)-structure if and only if it admits a GL(m; C)-structure. An Sp(m; R)-structure is called an almost symplectic structure or an almost Hamiltonian structure. If an almost symplectic structure is integrable with admissible coordinate system X', ..., x2 m so that

co= dx 1 A dxm+ 1 + • •• + dxm A dx2 m,

then d co =O. Conversely (see Appendix 1), if the form w defining an almost symplectic structure is closed, then co =dx 1 A dxm+ 1-+ • • • + dxm A dx 2 m for a suitable local coordinate system .X 1 9 ... 9 X2 m and the structure is integrable. An integrable almost symplectic structure is called a symplectic structure or a Hamiltonian structure. We observe that if an almost symplectic structure admits a torsionfree affine connection, then it is integrable. For the 2-form co defining an almost symplectic structure is parallel with respect to such a connection and hence is closed. (In cal-culating dco in terms of a local coordinate system, partial differentiation may be replaced by covariant differentiation when the connection is torsionfree, see for instance Kobayashi-Nomizu [1, vol. 1; p. 149]). A diffeomorphismf of M onto itself is an automorphism of the symplectic structure defined by a 2-form co if and only iff* co = co. Similarly, X is an infinitesimal automorphism if and only if Lx co= 0. An (infinitesimal) automorphism of a symplectic structure is called an ( infinitesimal) symplec tic transformation.

Set

CSp(m; R)= (AeGL(2m; R); tAJA=cJ, cat+) =Sp(m; R) x R -E,

csp(m; R)= Pleg1(2m; R); tAJ +JA=cJ, cell} =sp(m; R)+R.

where

{(240

gl(p; R)-F gl(q; R)={(A 0

GL(p; R) x GL(q ; R)=


A CSp (m; R)-structure is called a conformal-symplectic structure. For conformal-symplectic geometry, see Lee [1].

Example 2.9. G = GL (p, q; R) and g =g1(p, q; R), where GL(p, q; R) denotes the group of linear transformations of 1r, n = p + q, which leave the p-dimensional subspace RP defined by u"' • • • = u =0 invariant. In other words,

GL(p, q; R)=I(A B ) . AeGL(p; R), CeGL(q; R)} 0 C

gRA q; C R)= 1(A0 B );

Aegl(p; R), Cegl(q; R)},

where B denotes a matrix with p rows and q columns. Clearly, g contains an element of rank 1 and, hence, is of infinite type. The GL(p, q; R)-structures on M are in a natural one-to-one correspondence with the p-dimensional distributions on M, i. e., the fields of p-dimensional sub-spaces of tangent spaces. A GL(p, q; R)-structure is integrable if and only if there exists a local coordinate system x 1 , , x" such that a/ax l , aiaxP span the corresponding p-dimensional distribution. In other words, a GL(p, q; R)-structure is integrable if and only if the corresponding p-dimensional distribution is involutive, (see Frobenius theorem). An integrable GL(p, q; R)-structure is known as a foliation with p-dimen-sional leaves. If a GL (p, q ; 10-structure admits a torsionfree affine connec-tion, it is integrable. Indeed, if X and Yare vector fields belonging to the distribution, then the formula [X, /7] =V Y— Vy X (see Kobayashi-Nomizu [1; p. 133]) implies that [X, Y] also belongs to the distribution. Since an automorphism of a GL(p, q; R)-structure on M is a transfor-mation preserving the corresponding p-dimensional distribution, a vector field X on M is an infinitesimal automorphism if and only if, for every vector field Y belonging to the distribution, [X, Y] belongs to the distribu-tion.

Example 2.10. G = GL(p ; R) x GL (q ; R) and g = gl(p; R)+ gl(q; R), p + q =n. In other words,.

Bo) ; AEGL(p; R), BeGL(q; R)},

BO); Aegl(p; R), Begl(q; R)}.

Clearly, g contains an element of rank 1 and, hence, is of infinite type. The GL(p; R) x GL(q; R)-structures are in a natural one-to-one corre-spondence with the set of pairs (S, S'), where S and S' are complementary distributions of dimensions p and q respectively. A GL(p; R) x GL(q; R)-

3. Two Theorems on Differentiable Transformation Groups 13

structure is integrable if and only if the corresponding distributions S and S' are both involutive, that is, there exists a local coordinate system x', , xn such that a/axi , , 0/axP span S and a/axP+ 1, , alaxn span S'.

Example 2.11. G= {1) and g =O. The {1 } -structures on M are in a natural one-to-one correspondence with the fields of linear frames over M. A manifold M is said to be parallelisable if it admits a (1)-structure. The automorphism group of a (1)-structure will be studied in the next section (Theorem 3.2).

3. Two Theorems on Differentiable Transformation Groups

The theorems in this section will allow us to prove that the automorphism groups of many geometric structures are Lie groups.

Theorem 3.1. Let (1/46 be a group of differentiable transformations of a manifold M. Let S be the set of all vector fields X on M which generate global 1-parameter groups (p i = exp t X of transformations of M such that (A0:5. If the set S generates a finite-dimensional Lie algebra of vector

fields on M, then is a Lie transformation group and S is the Lie algebra of IA

Proof Let g* be the Lie algebra of vector fields on M generated by S. Let a be the connected, simply connected Lie group with Lie algebra g*; it is an abstract Lie group and is not a transformation group. For each element X of g*, we denote by et X the 1-parameter subgroup of a generated by X while we denote by exp t X the 1-parameter local group of local transformations of M generated by the vector field X. Then the group a acts locally on M in the following sense. There exist a neighbor-hood U of {I} x M in x M and a mappingf: U --FM such that

f (et X, p) = (exp t X)p for (et x, p)e U x M.

Lemma 1. Given X, Ye g*, we define Ze g* Z= (ad ex) Y If X, Y are in S, so is Z.

Proof of Lemma 1. From et z = ex e1 we obtain

(exp t Z) p = (exp X) (exp t Y) (exp — X) p

If X, YeS, then the right hand side is defined for all p and t. Hence, (exp t Z)p is also defined for all p and t. This implies that Z is in S.

Lemma 2. S spans g* as a vector space.

Proof of Lemma 2. Let 1/ be the vector subspace of g* spanned by S. By Lemma 1, we have (ad es) S S and, hence, (ad es) V c V Since S

by e tz =ex ety e- x, i.e.,


generates g*, es generates 6. Hence, (ad () V c V. In particular, (ad el') • V = V, which implies [V, V] c V

Lemma 3. S = g*.

Proof of Lemma 3. Let X1 , . , X,. E S be a basis for g*. Then the mapping

ai Xi eg*

xi er xr e

gives a diffeomorphism of a neighborhood N of 0 in g* onto a neighbor-hood U of the identity element in ft. Let YE g*. Let be a positive number such that et 1' e U for I tI <(5. Then, for each t with I tI <6, there exists a unique element E a (t) X i EN such that

et Y = eat (0 X1 ear(OXr .

The action of exp t Yon M is therefore given by

(exp t Y) p = (exp a1 (t) X 1 ) . . . (exp a,. (t) Xr) p for p e M and I t I <0 .

This shows that every element Y of g* generates a global 1-parameter group of transformations of M. Hence, YES, thus completing the proof of Lemma 3.

Let 0* be the Lie transformation group acting on M generated by g*; 6* exists since every element of g* generates a global 1-parameter group of transformations of M by Lemma 3. Since 15* is connected, the assump-tion in the statement of Theorem 3.1 implies /5* c 15. Let cpE15 and be a 1-parameter subgroup of 05*. Then cp (p - 1 is a 1-parameter group of transformations of M contained in 6. From the contruction of 15* it follows that this 1-parameter group is a subgroup of 15*• Since 15* is generated by its 1-parameter subgroups, this implies that 15* is a normal subgroup of 15. Each Tee, defines an automorphism A0 : 0* -->15* by A0 (0)=T cp'. Since the automorphism A0 sends every 1-parameter subgroup of 05* into a 1-parameter subgroup of /5*, it is continuous (see Chevalley [1; p. 128]).

Lemma 4. Let 0 be a group and (5* a topological group contained in 0 as a normal subgroup. If A 49 : (§* 1:6* is continuous for each cp e %, then there exists a unique topology on 0 which makes 0* open in 0.

Proof of Lemma 4. If { V) is the system of open neighborhoods of the identity element in 0*, we take (4) (V)) as the system of open neighbor-hoods of cp E15 in 15. It is a trivial matter to verify that 15* is open in 15 with respect to the topology thus defined in G. The uniqueness of such a topology is also evident.

Applying Lemma 4 to our case, we see that 15 is a topological group with identity component 15*. Since 0* is a Lie group, its dif-

3. Two Theorems on Differentiable Transformation Groups 1.5

ferentiable structure can be translated to other connected components of 0. The differentiability of the action 0* x M --*M implies that of

x q. e. d.

As we shall see in §5 (Theorem 5.1) the study of the automorphism group of a G-structure can be reduced to the case of a {1}-structure if the Lie algebra g is of finite type. The following theorem is therefore basic.

Theorem 3.2. Let M be a manifold with a (1).-structure (j. e., an absolute parallelism). Let 91 be the group of automorphisms of the {1}-structure. Then 9.1 is a Lie transformation group such that dim 91 dim M. More precisely, for any point pe M, the mapping ae9.1-- a(p)eM is injective and its image {a(p); ac U} is a closed submanifold of M. The submanifold structure on this image makes 91 into a Lie transform'ation group.

Proof Let el , ..., en be everywhere linearly independent vector fields on M defining the given {1}-structure. Let V be the set of vector fields v which are linear combinations of el , e„ with constant coefficients. Then V is a vector space of dimension n. By definition, 91 consists of transformations a of M which leave each ye V invariant. In other words,

(*)

a 0 exp v =(exp v)0 a for ae91, ve V,

wherever exp v is defined. (In general, (exp t v) p is defined only for small values of t depending upon the point pe

Lemma 1. The mapping a e 91—* a(p)eM is injective.

Proof of Lemma 1. Let Fa be the fixed point set of a e 91. It is a closed subset of M. If q e Fa , then the set of points (exp v) q covers a neighborhood of q when v varies in a neighborhood of the origin in V. Hence, the equality (*) implies that this neighborhood of q is in F. Since Fa is closed and open, either Fa =A4 so that a is the identity element or Fa is empty.

Lemma 2. The set {a (p); aell) is closed in M for each pe M.

Proof of Lemma 2. Let {ak } be a sequence of elements of 91 such that ak (p)--* q for some qe M. We want to construct an element a of 91 such that a (p). q. We define the transformation a first in a neighborhood of p by setting

a((exp v)p).(exp v) q

for all v which are in a neighborhood of the origin in V so that both (exp v) p and (exp v) q are defined. Then a(p)=1im a k (p') for all p' in a neighborhood of p for which the transformation a is thus defined. Using (*) we extend the definition of a along curves from p. Since a (p') = lim a k (p'), the extended map a is independent of the choice of curves. From the

16 ' L Automorphisms of G-Structures

construction of a, it is clear that a is a local diffeomorphism. To see that a-1 exists, we observe first that ak-1 (q)--* p and then apply the same con-struction to obtain a -1 as the limits of (an. It is easy to see that a is an automorphism of the (1)-structure.

Let I be the set of vector fields X on M such that [X, y] =0 for all y in V It is a Lie algebra of vector fields.

Lemma 3. For each point peM, the restriction map Xel--* Xp eTp (M) is infective. In particular, dim I dim M.

Proof of Lemma 3. This is immediate from (*).

Let a be the set of vector fields X el which generate a global 1-para-meter group of transformations of M. The Lie algebra of vector fields generated by the set a is contained in I and hence is of finite dimension. We apply Theorem 3.1 to (5= 91 and S =a. Then we can conclude that a is a Lie algebra and 91 is a Lie transformation group with Lie algebra a. Since the action 91 x M --*M is differentiable and since the image of the injection ae21 a(p)e M is closed, {a (A; ae 91) is a closed submanifold of M and the mapping

ae9.1--*a(p)e(a(p); ae91) is a diffeomorphism. q.e.d.

Theorem 3.1 is due to Palais [1]; the proof given here is from Chu-Kobayashi [I]. Theorem 3.2 is due to Kobayashi [I]; the original proof was closer to that of Myers-Steenrod [1] for the group of isometries of a Riemannian manifold.

Another important criterion for a topological transformation group to be a Lie transformation group is the following theorem of Bochner-Montgomery [1] we state without proof (see also Montgomery-Zip-pin [1]).

Theorem 3.3. Let 0 be a locally compact group of differentiable trans-formations of a manifold M. Then 0 is a Lie transformation group.

4. Automorphisms of Compact Elliptic Structures

We recall that a linear Lie algebra g g! (n; R) is said to be elliptic if g contains no matrix of rank I (see §1) and that gl(m; C), as a subalgebra of gl(2 m; R), is elliptic (see Example 2.4). The purpose of this section is to prove

Theorem 4.1. Let P be a G-structure on an n-dimensional compact manifold M. If g is elliptic, then the group of automorphisms of P is a Lie transformation group.

4. Automorphisms of Compact Elliptic Structures 17

Proof We shall show that the Lie algebra of infinitesimal automor-phisms of P is finite dimensional. Then the theorem will follow directly from Theorem 3.1 (We remark that the proof of Theorem 3.1 becomes extremely simple when M is compact because every vector field on M generates a global 1-parameter group of transformations.)

The essential idea is to construct a system of elliptic partial differential equations of which the infinitesimal automorphisms of P are solutions.

Since g is a linear subspace of gl(n; R), it may be defined by

g={(cii)egl(n; R); E cf a =0 for 2=1, , N ,

where the 42 are constants and N is the codimension of g in gl(n; R). Let VI , , V„ be vector fields locally defined on M which define a

local cross section of P. Let 0, ..., co" be the dual basis of VI , ..., V.; they are linearly independent 1-forms such that d(17;)=6Li . Let X be an infinitesimal automorphism of P and write

X=E Vi .

Fixing a connection in P, denote by V its covariant differentiation operator. Then write

VX =E eocoi

(The coefficients Vo are defined by the equality above.) Let xl , x" be a local coordinate system in M. Then

k (1) = E axk Ai(x)+•1

where (A .1) is defined by E wi=dac k and the dots indicate the terms not involving partial derivatives of X i.

In the definition of the torsion tensor

T(X, Vi)=V x VI, — [X, we have

(2) vx E Vj with (p.i(x)) e g

and

(3) [X, Vi] =E vi(x) V; with (vii (X)) E g

While (2) follows from the fact that P is invariant under parallel displacement and VI , V„ is a frame field belonging to P, (3) follows from the fact that exp(t X) is a 1-parameter group of automorphisms of P. On the


other hand, we can write

(4) Vvi X=EVA and

(5) T(X, Vd= E TA vk where Thi are defined by

T(V» VJ =TJ Vk. From

Vv, X + Vx [X, we obtain

(6) E Tki = •

Since (pi— vi) belongs to the Lie algebra g, (6) implies

for 2=1,..., N .

Substituting (-1) into this, we obtain

a (7) Ebil A T9-7-xk +••• =0 with bil 2 =E4 2 ,411,

where the dots indicate terms not involving partial derivatives of e. Differentiating (7) with respect to xh and multiplying by b, 2 , we obtain

a2 E bhm2 b.1 2 axkaxh + =0,

h, j, k,

where the dots indicate terms involving partial derivatives of order 1 or less. We shall show that (8) is a system of elliptic partial differential equations. We consider the following symbol of (8), which is an n x n matrix:

E bhnt b.1; Vk VOi t n h, k,

where v= (y1 , vn) is an arbitrary nonzero covector. The problem is to show that this matrix is non-singular. Let s = (s', . . . , s") be a vector. We want to show that if

(9) E bmh Vk Vh S j = 0

h, Lk,

then s= O. Multiply (9) by sm. Sumrriing over the index m, we obtain

(10) t =0, where ta = >b2 vk

(8)

j,k

5. Prolongations of G-Structures 19

Hence, t2 =0, i. e.,

(11) E c4 vk si=0 for 1=1, ..., N . i,j,k

Hence, the matrix (E Ail vk s 1) ,, belongs to the Lie algebra g. This k

matrix is the product of two matrices (An and (Vic si). Since (All) is non-singular and (vk si) is of rank 1 if s*0 and since g contains no matrix of rank 1, it follows that s =O.

We have established that the infinitesimal automorphisms X of P satisfy a system of elliptic partial differential equations (of second order) (8). It follows (see, for instance, Bochner [3]) that the Lie algebra of infinitesimal automorphisms X is finite dimensional. (See also Douglis-Nirenberg [1]. If we choose any Riemannian metric on M and denote by X', X", X" the first, second and third covariant derivatives of X, then the Lie algebra of infinitesimal automorphisms X of P is a Banach space with norm 11X11 defined by

II X II = Max 1X1 + Max 1 X'l + Max IX"I + Max I X m l. peM peM peM peM

From a theorem of Douglis and Nirenberg, it follows that this Banach space is locally compact and hence is finite dimensional, (see Ruh [1] for more details)). q.e.d.

Corollary 4.2. The automorphism group of a compact almost complex manifold is a Lie transformation group.

Corollary 4.2 was proved by Boothby-Kobayashi-Wang [1] in the same manner as Theorem 4.1. Its generalization, Theorem 4.1, is due to Ochiai [2]. In the locally flat case, Theorem 4.1 was proved by Guillemin-Sternberg [1]. Ruh [1] also proved a similar result. As in Ochiai [2], Theorem 4.1 can be proved without the aid of a connection in P.

5. Prolongations of G-Structures

Let V=R" and G be a Lie group of linear transformations of V We recall (see § 1) that the first prolongation g i of the Lie algebra g of G is the space of symmetric bilinear mappings t: V x V—> V such that, for each fixed vi e V, the mapping ve V—) t(v, vi)e V is in g. We define now the first prolongation G 1 of G to be the group consisting of those linear transformations i of V+ g induced by the elements t of gi as follows:

i(v)= v + te , v) for VE V,

i(X) = X for xeg.

Symbolically, i is a matrix of the form

i = (4, 0) $

t Ir


where r is the dimension of g. Then G1 is a vector group isomorphic to gl . We recall (see § 1) that the k-th prolongation gk of g is the space of

symmetric multilinear mappings

t : V x • • • x T7,— V ....—„.—..

k+ 1-times

such that, for each fixed v1 , ... , yk e V, the linear transformation ye V—* t(r, th, ..., vk)e V is in g. The k-th prolongation Gk of G is the group consisting of those linear transformations 1 of V+ g + g1 + • • • + gk _ 1 induced by the elements t of gk as follows:

i(y)=v+t(•, ...,•, v) for yeV,

t(x)= x for xeV+g+g1+•••+gk-1.

Symbolically, I is a matrix of the form

(4, 0 0

1.

i = 0 IN 0),

t 0 1

where N = dim(g + g 1 + • - • + gk _ 2) and r = dim gk _ 1 . Then Gk is a vector group isomorphic to gk .

It is easily seen that the first prolongation of Gk_i coincides with Gk.

Thus Gk can be obtained from G by successive first prolongations. Denote the dual space of Vby V*. Then V® A2 V* may be considered

as the space of all skew-symmetric bilinear mappings from V x V into V, and similarly, g 0 V* may be identified with the space of linear mappings from V into g. Define a linear mapping a: g® V* —> VOA 2 V* by

(af)(v 1 ,v 2)=— f (v 2 ) y 1 + f (v4) y 2 for fe g 0 V*, y1 , v2 e V.

It is clear that f is in the kernel of a if and only if the mapping (y1 , y 2 )e V x V —> f (y i) v2 e V is in the first prolongation gl .

We choose once and for all a linear subspace C of VOA 2 V* such that

VOA V*=0(ge V*)+ C.

In general there is no natural way of choosing C. Let P be a G-structure on an n-dimensional manifold M; it is a

subbundle of L(M) with structure group G. Let 0 be the canonical form on P; it is a V-valued 1-form on P An n-dimensional subspace H of the

5. Prolongations of G-Structures 21

tangent space 7: (P) at u is said to be horizontal if 0: H—> V is an isomor-phism. The exterior derivative d0 evaluated at ue P is a skew-symmetric bilinear mapping (An : T(P) x 7:(P)—> V. In view of the isomorphism 0: H—> V, d0 restricted to H x H defines a skew-symmetric bilinear mapping V x V—* V, i.e., an element c(u, H)E VOA 2 V*, called the torsion corresponding to (u, H).

To discuss the dependence of e(u, II) on H, we fix a pair (u, H) and choose a connection form a.) on P such that H is horizontal with respect to co, i. e., co (X) =0 for X e H. The so-called first structure equation reads

d0= —co A0+0,

where 0 is the torsion form. Let y1 , v2 E V and X1 , X2E11 such that 0(Xi)= vi , i =1, 2. Then c(u, H)(vi , v2)= 0(X1 , X2 ), thus justifying the name "torsion" for c(u, H). Let H' be another horizontal subspace of T.(P) and Xi , X'2 e H' be vectors such that 0 (X) = v i , i =1, 2. Then, since eo (X;, r2)= e(x1, X2), we obtain

c (u, 11)(v1 , v2) —c(u, H)(v i , v 2)= dO(Xi, X;)—d0(X 1 , X2 )

= —(w A û)(X, X'2)+ (co A 0)(X 1 , X2 )

= — co (Xi) 0 (r2 )+ co (X) 0 (Xi) .

If we define an element f of g 0 V*, j. e., a linear mapping from V into g by

f (v) = co(r) v e V,

where XiEH' is determined by 0(X')= v, then

— co (Xi) 0 (X'2 ) + co (X) 0 (Xi) = — (a f)(v il v 2 ) .

This shows that c (u, H')— e(u, H) is an element of a (g 0 V*). Each horizontal subspace H of T. (P) determines a linear frame of

the manifold P at u as follows. Since the structure group G acts on P, every element A of g induces a vertical vector field A* on P, called the fundamental vector field corresponding to A. Hence we have a linear mapping g —* Tii (P) which sends A into A,*, . On the other hand, we have a linear mapping V--* H c T„(P) which sends v into the vector XEH defined by 6(X)= v. Adding these two linear mappings, we obtain a linear isomorphism V+ g —* 7, (P), which is by definition the linear frame of P determined by H. If we take a basis el , ..., en, in V+g in such a way that el , ... , en is the natural basis for V= It" and en .4_ 1 , ... , en +,. is a basis for g, then the image of this basis under the isomorphism V+ g —* T(P) is the linear frame determined by H (if one wants to define a linear frame at u as an ordered basis of 7, (P) rather than a linear isomorphism from a fixed vector space V+ g onto T. (P)).


We are now in a position to define the first prolongation R of a G-structure P over M. The definition will depend on the fixed subspace C of VOA' 17* complementary to (3(g OP). Let 13 be the set of linear frames over P corresponding to horizontal subspaces H OE 7;,(P) such that ue P and c(u, Me C. Then P1 is a Grstructure over P. In fact, if a e GL(n + r; R) and z E 13 = L(P), then z • a is defined by

z • a: I/-Fg—g— 17+ g !_) TL(P), and z • a is in Pi if and only if ae . (To see this, let H and H' be the horizontal subspaces of T. (P) corresponding to z and z • a, respectively. By definition, c(u, H) is in C. We know that c(u, H') — c(u, H) is in a(g Pi). Since z • a is in /3 if and only if c(u, H') is in C, it follows that z • a is in P1 if and only if c(u, H') — c(u, 11)e C na(g V*)=0. This means that z • a is in Pi if and only if the element f of g(:)1/* defined above is in the kernel of a. But the kernel of a is precisely gl . Our assertion is now immediate.)

The k-th prolongation Pk of P is defined inductively by Pk = (Pk. J = the first prolongation of Pk_ 1; it is a Gk-structure over Pk_ 1 .

Let (p be an automorphism of a G-structure P over M; it is a trans-formation of M such that the induced bundle automorphism (19* of L(M) leaves P invariant. We denote the restriction of ([4 to P by Pi. Then (p i

is an automorphism of the Grstructure R over P To see this, let H be a horizontal subspace of 7(P) such that c(u, H)e C so that the correspond-ing linear frame z of P at u is in R. From the fact that 0 is invariant by

it follows that c ((pi (u), (Pi* (H)) = c(u, Me C. Hence, the linear frame (p i* (z) corresponding to the horizontal subspace (p i* (11) is in P1 . This proves our assertion.

We can construct inductively a transformation (p k of IL .1 which is an automorphism of the Gk-structure Pk over 11_ 1 .

Theorem 5.1. Let P be a G-structure on an n-dimensional manifold M and 91 the group of automorphisms of P if the Lie algebra g c g! (n; R) is of finite type of order k, then 91 is a Lie transformation group of dimension -dim(17-Fg+g 1

Proof Since gk =0, Gk consists of the identity element only and the Gk-structure I over /1„, is a (1)-structure, i.e., an absolute parallelism on 11„. 1 . Let 21. denote the group of automorphisms of the Gi-structure /3 over n_ i , i. e., transformations of inducing automorphisms of R. By Theorem 3.2, 9Ik is a Lie transformation group of dimension dim for a fixed element z of R_ 1 , the mapping (/e2tk —/J (z)ePk_1 imbeds 9Ik as a closed submanifold of Pk _1. On the other hand, we can imbed 21 into 91k as a closed subset by mapping (pe21 into (pk e 91k . Hence, 91 is a Lie transformation group of dimension dim 11_ 1 . Since dim Pk_ 1 = dim (17+ g + + • • • + g-j, the theorem is now proved. q. e. d.

+ • • + gk_i).

6. Volume Elements and Symplectic Structures 23

Theorem 5.1 was first proved in special cases, e. g., for Riemannian conformal, projective structures and, more generally, for Cartan con-nections as an application of Theorem 3.2 (Kobayashi [1, 4]), for pseudogroup structures of finite type and hence for integrable G-struc-tures of finite type (Ehresmann [4], Libermann [3]) and then in this general form by Ruh [1] and Sternberg [1].

6. Volume Elements and Symplectic Structures

We shall reconsider some of the examples discussed in § 2. Since we have established basic theorems on automorphism groups for elliptic G-structures and for G-structures of finite type in §§ 4 and 5, we shall be concerned with G-structures of non-elliptic infinite type in this section.

Let M be an n-dimensional manifold and consider the GL(n; R)- structure on M, i. e., the bundle L(M) of linear frames over M (see Example 2.1). The group of automorphisms of the GL(n; R)-structure is nothing but the group of diffeomorphisms of M, which will be denoted by Z(M). Similarly, the Lie algebra of infinitesimal automorphisms of the GL(n; R)-structure is the Lie algebra I(W) of vector fields on M. When M is noncompact, practically nothing seems to be known about Z(M) and 1(M). For instance, a natural question would be whether to(M) can be made into an infinite dimensional Lie group in a suitable sence. One of the difficulties seems to be lack of the corresponding Lie algebra. Since some vector fields are not complete, i. e., cannot be inte-grated globally, 1(M) is too large to be the Lie algebra of Z(M). On the other hand, the subset of 1(M) consisting of complete vector fields is not even a linear subspace of 1(M) (see Palais [1] for an example of two complete vector fields whose sum is not complete). Leslie [1] has shown that if M is compact, then Z(M) can be made into a Fréchet Lie group. Perhaps more useful is a strong ILH-Lie group ,structure introduced in Z(M) by Omori [1] in the case when M is compact (where ILH stands for "Inverse Limit of Hilbert"), see also Ebin-Marsden [1]. If one wishes to generalize Omori's results to the case of a noncompact manifold M, then the group to be considered is perhaps an appropriate completion toc (M) of the subgroup 1),(M) of Z(M) consisting of diffeo-morphisms with compact support, i. e., transformations which act trivi-ally outside compact sets, with Lie algebra Ic (M) consisting of vector fields with compact support. Clearly, every element of X(M) is a com-plete vector field and generates a 1-parameter subgroup of 1%.(M). The Lie algebra of t(M) would consist of vector fields decreasing rapidly at infinity in a suitable sense.

We shall now consider an SL(n ; R)-structure on M, i. e., a volume element it on M (see Example 2.3). Let 91(M, p.) (resp. a(M, p.)) denote


the group (resp. the Lie algebra) of transformations f (resp. infinitesimal transformations X) such that f*t = (resp. Lx =0). If M is compact, then 9I(M, it) is a closed Fréchet Lie subgroup and hence a closed strong ILH-Lie subgroup of Z(M) with Lie algebra a (M, it) (Omori [2], Ebin-Marsden [1]). Again it is possible that the correct group to be considered for a noncompact M is a completion 9f,(M, y) of the group 91,(M, p)=9.1(M, /.4) 1),(M) of ti-preserving transformations with com-pact support with the corresponding Lie algebra ac (M, ;4= a(M, p)r)

k(M)- The following result is due to Boothby [3].

Theorem 6.1. Given a volume element it on a manifold M of dimension n 2, the group 91,(M, it) of /4-preserving transformations with compact support (in fact, already the subgroup generated by ac (M, 14)) is k-fold transitive on M for every positive integer k.

We recall that a group acting on M is said to be k-fold transitive if for arbitrary two sets of k distinct points {p 1 , , pk } and {q 1 , , qk ) there is an element of the group which sends pi into q i for all i =1, . , k. Follow-ing Boothby we say that a group acting on M is strongly locally transitive if for each point peM and each neighborhood U of p there are relatively compact neighborhoods V and W of p with Wc U, and for each q e W there is an element of the group which leaves fixed every point outside V and sends p into sq.

Proof. The following general lemma will be used again later.

Lemma 1. If a group 6 is strongly, locally transitive on M, it is k-fold transitive on M for every positive integer k.

Proof of Lemma 1. Let {pi , , p lc } and {q 1 , qk } be two sets of k distinct points. For each i, let ci be a curve from p i to q i chosen in such a way that e1 , , ck are mutually disjoint. For each i, let Ni be a neighbor-hood of ci chosen in such a way that N1 , , N disjoint. It suffices to show that for each i there is an element gi of 6 which sends pi into q i and leaves every point outside Ni fixed. To obtain such an element gi , for each point r on ei we choose relatively compact neighborhoods V,. and W,. of r with W,. n. -17,..u=Ni in the manner described in the definition of "strongly locally transitive action". Then we choose points p 1 =r0 , 11, , rm = q i on ei in such a way that ri e Wri _ i for j= 1, , m. Then we can send pi successively to r1 , , r„, q 1 by elements of 6 without disturbing the points outside Ni .

Lemma 2. Let it be a volume element on M. Then the group generated by ac (M, p.) is strongly locally transitive on M.


Proof of Lemma 2. Given a point p e M and a neighborhood U of p, let x', be a local coordinate system around p such that p.= dx 1 A -•• A dxn; such a coordinate system exists since every SL(n ; R)-structure is integrable (see Example 2.3). Let V and W be neighborhoods of p with Wc V c V U defined by

V: ix i l<a and W: lxi l <b, where 0<b<a.

Let q e W. Applying a linear change to the coordinate system, we may assume that q has coordinates (c, 0, , 0). Let f be a function with support in V such that f depends only on the variables x', x2 and f =x2 on W. Then define a vector field X by

a af a x = ax2 axi axi ax2 - It is then easy to verify that X is a it-preserving infinitesimal transfor-mation with support in V and its orbit through the origin (0, 0, , 0) passes through the point q.

Now the theorem follows immediately from Lemmas 1 and 2. q. e. d.

We mention a theorem of Moser [1] which says that on a compact manifold all volume elements are essentially equivalent. This has been used in Omori's work mentioned above.

Theorem 6.2. Let p. and y be two volume elements on a compact manifold M. Then there is a transformation f of M such that f* 1.1.=y if and only if f /.4 = j v.

In connection with possible generalizations of Omori's results to the noncompact case, we should point out that Theorem 6.2 can be general-ized to a noncompact M as follows. Let and y be two volume elements on M. Then there is a transformation fEZ,(M) with compact support such that f* g=y if and only if there is a compact subset K of M such that S /4= 5v and it=y outside K. K K

We shall now consider a symplectic structure on a manifold M of dimension n = 2m, i. e., a closed 2-form co of maximal rank (see Example 2.8). A transformation f (resp. an infinitesimal transformation X) of M is said to be symplectic if f*w =co (resp. Lx co =0). We denote by 91(M, co) (resp. a (M, co)) the group (resp. the Lie algebra) of symplectic transformations (resp. infinitesimal symplectic transformations). If M is compact, then 9I(M, co) is a closed Fréchet Lie subgroup and hence a closed strong ILH-Lie subgroup of to (M) with Lie algebra a (M, co) (Ebin-Marsden [1], Omori [2], Weinstein [2]). Again the question arises whether the group 91,(M, w)= 9.1[(M, co) n 1),(M) of symplectic transformations with corn-


pact support is a strong ILH-Lie group with the corresponding Lie algebra a,(M, co)= a (M, w) r' X,(M) of infmitesimal symplectic transfor-mations with compact support.

The following result is due to Libermann [3].

Theorem 6.3. Let M be a symplec tic manifold with closed 2-form co of maximal rank. Under the linear isomorphism between the space 3E(M) of vector fields and the space Jar (M) of 1-forms given by X EX(111)— —ix coese(M), the space a(M, co) of infinitesimal symplectic trans-

formations is isomorphic to the space (M) of closed 1-forms. Under the same isomorphism, the derived subalgebra [a(M, co), a(M, co)] is mapped into the space al (M) of exact 1-forms.

Proof. We apply the formula Lx =do i x + ix od to co (see for example Kobayashi-Nomizu [1; vol. 1, p. 35]). Since co is closed, we obtain

Lx co=dot x co.

This shows that X is an infinitesimal symplectic transformation if and only if ix co is closed. To prove the second statement, we use the formula 1 [X, yi= [Lx ,t y] (see Kobayashi-Nomizu [1; vol. 1, p. 35]). Assume that X and Y are infinitesimal symplectic transformations. Since Lx (0=0 and doc y w = Ly w =0, we obtain

/my] w=L x 0 t y w— t y 0Lx w=do tel y co= 2d(co(Y, X)).

This proves the second assertion. q. e. d.

This is probably an appropriate place to mention the classical Poisson bracket (f, g). Let f and g be two functions on a symplectic manifold M. Let Xf and Xg be the vector fields corresponding to the exact 1-forms df and dg, respectively, under the duality defined by the symplectic form co, i.e.,

df= i xf a), dg= — t x,co.

(f, g) = ix.° txf co (= 2 co (Xf , Xg))

It is easy to verify that the space ,F(M) of functions on M with Poisson bracket { } is a Lie algebra. From the last formula in the proof of Theorem 6.3, it follows that d{f, g) corresponds to [Xf , Xg], i.e., d (f, g) = — t [x xgj co. This fact implies that the mapping f Xf defines a Lie algebra homomorphism from .F(M) into a (M, co). The kernel of this homomorphism consists of the constant functions. If we express co in terms of an admissible coordinate system x', , x2 " as

co =dx 1 dxm+ 1 +--- +dx" dx 2 m

We set


(see Example 2.8), then by a simple calculation we obtain the classical formula

m af ag af ag {f1g)= E axi axm+i axos-Ei axi . il

The following result, proved first by Hatakeyama [1] in the compact case, is due to Boothby [3].

Theorem 6.4. Given a symplectic manifold M with closed 2-form co of maximal rank, the group 9-UM, co) of symplectic transformations with compact support (in fact, already the subgroup generated by o, (M, co)) is k-fold transitive on M for every positive integer k.

Proof In view of Lemma 1 in the proof of Theorem 6.1, it suffices to prove that the group generated by ct,(M, co) is strongly locally transitive on M. Given a point peM and a neighborhood U of p, let x1 , ..., x'm be a local coordinate system around p such that co = dx1 A d xm+ 1 + • • • + dxm A dx2 m (see Example 2.8). Let 17 and W be neighborhoods of p with Wc Tic .V" u defined by

17: lxi l< a and W: IA < b, where 0<b<a.

Let qe W Applying a linear symplectic change to the coordinate system, we may assume that q has coordinates (e, 0, ... , 0). Let f be a function with support in 17 such that f= — xm+ 1 on W. Then the infinitesimal symplectic transformation X1 defined by tx, co= df has support in 17

and coincides with 0/Ox' on W. Hence, the 1-parameter group generated by X1 maps the origin p= (0, 0, ... , 0) into q =(c, 0, ... , 0). q. e. d.

A result similar to Theorem 6.2 on the symplectic forms on a compact manifold is known but it is not as strong as Theorem 6.2, see Moser [1].

Given a symplectic structure on M with closed 2-form co of maximal rank, a transformation f of M is said to be conformal-symplectic if f*co=cp•co, where cp is a function on M. Clearly, f is conformal-symplectic if and only if it is an automorphism of the CSp(m; R)- structure defined by co (see Example 2.8). Since co and f* co are closed, we obtain dcp A co =O. Making use of the expression co = dx1 A dxm + 1 + --- + dxm A dx2 m, we can easily conclude that dcp =0 if m2. Hence (Liber-mann [3]),

Theorem 6.5. If f is a conformal-symplectic transformation of a symplectic manifold M of dimension 2m . 4 with closed 2-form co of maximal rank, then

f* (0=c • co,

where c is a nonzero constant. If M is compact, then c= +1.


Proof. The last assertion follows from

S f* (QM= ± S

where the sign is positive or negative according as f is orientation-preserving br reversing, q. e. d.

• An example of a symplectic manifold is a Kahler manifold with

Kahler 2-form co (whose complex structure and Riemannian structure are forgotten).

Another example is provided by the cotangent bundle T* (M) of any manifold M with the natural symplectic form co defined as follows. Let y be the 1-form on T* (M) defined by

y (X) = (ir* X) for X e (T* (M)) ,

where n: T* M is the natural projection so that n * X e 7;4 (M). Let x', xi' be a local coordinate system in M and x', xn, pi , be the induced local coordinate system in T* (M), i.e., x 1 () =x'(7E and p = (alaxi) for e T* (M). Then

y =Ept dx i

co=dy =Edpi A dx L.

For more results on automorphisms of (almost) symplectic and conformal-symplectic structures, see Lefevre [1-3], Libermann [2], Lichnerowicz [4]. On characterization of the symplectic structure on the cotangent bundle T*(M), see Nagano [11 ].

7. Contact Structures

Let M be a manifold of odd dimension 2m +1. By a contact form on M we mean an open cover Wi ) of M together with a system of 1-forms {y,} such that

(1) each y i is a 1-form defined on Ut of maximal rank in the sense that yi A (dyi)tm *0 everywhere on Ui ,

(2) we have y, =f yi on U, U,, where fii is a function on Ui n Ui (without zeros).

Two such forms { U , y i ) and {VA , 6A ) are said to be equivalent if yi =hiA (5 1 on ui n VA, where hiA is a function on ui 1/1 (without zeros). An equivalence class of contact forms is called a contact structure. For

Set

7. Contact Structures 29

simplicity's sake, we say "a contact structure (Ui , y e )" instead of "a contact structure represented by ( Ui , y e )".

Given a contact structure (ui , y e), we define a vector subbundle of rank 2m of the tangent bundle T(M) by setting

Ex = (X e Tx (M); y e (X)=0) if x e ui . •

From condition (1) it follows that (dy e)" *0 on the fibre Ex and hence dy e defines a non-degenerate skew-symmetric bilinear form on E. This bilinear form on Ex is defined uniquely up to a nonzero constant multiple. It follows that the vector bundle E is orientable.

Let L be the quotient line bundle T(M)/ E. Each contact form ((A, y e ) gives rise to a globally defined 1-form with values in the line bundle L in the following manner. Since yi annihilates Ex , it can be considered as a linear functional on Tx (M)/ Ex . Hence, the equation

yi(se)= 1

defines a cross section of L over ui . Since y e (se) = 1 =Ms.), we obtain

si =fe; se on Ui n 1./.1 .

It follows that the form 5; defined by

5" =Yi si

is a globally defined 1-form with values in L. It is easy to verify that an equivalent contact form gives rise to the same L-valued 1-form 5). Thus 5/ depends only on the contact structure defined by (ui , yi).

Given a contact structure {ui , y) on M, a transformation f of M is called a contact transformation if (f 1 ui , f* y e ) and ( U;, y e ) are equivalent. Consequently, an infinitesimal contact transformation X is defined by the condition

Lx y e = ge • y e (where ge is a function on Ue).

More geometrically, f is a contact transformation if and only if the induced bundle automorphism f* : T(M)—* T(M) sends the subbundle E into itself, that is, if and only if f is an automorphism of the GL(n —1, 1; R)-structure on M defined by the subbundle E (see Example 2.9).

Writing y for a contact structure (Ue , y e ) for simplicity's sake, we denote the group of contact transformations by 91(M, y) and the Lie algebra of infmitesimal contact transformations by a(M, y). If M is com-pact, 91(M, y) is an ILH-Lie group, Omori [3]. As in the preceding section, we denote the subgroup (resp. subalgebra) of 91(M, y) (resp. a(M, y)) consisting of elements with compact support by 91, (M, y) (resp. ac (M, y)).

30 1. Automorphisms of G-Structure

Theorem 7.1. Given a contact structure {Ui ,y i} on M, let 5; be the corresponding 1-form on M with values in the real line bundle L =T(M)/E, where E is the subbundle of T(M) defined by y i = O. Then the mapping X e a(M, y) —*5; (X)e H° (M; L) gives a linear isomorphism from the space a(M, y) of infinitesimal contact transformations onto the space H° (M ; L) of cross sections of the line bundle L over M.

Proof To prove that the mapping is injective, let X ea(M, y) and assume . /'(X)=0, i. e., y i (X)= 0. Then

lye:4h= ixOdYi±do ixh=LxVi=gi•Yi.

Let Yek OE Tx (M) at xE (A. Then

iy oix odyi =gi • (t y y i)=0.

Since X is in E by our assumption y 1 (X)=0 and since dyi defines a non-degenerate bilinear form on Ex , we may conclude that X =0. To prove that the mapping is surjective, let s be a cross section of L. As before, let si be the cross section of L over Ui defined by y i (si)= 1 and write

s= hi si ,

where hi is a function on Ui . Let Si be the vector field on Ui defined by

yi (S i). 1 and ts , 0 dy i =O.

The projection T(M) —> L=T(M)/E maps Si into si . The vector field X we are looking for must be of the form

X =h i Si + Yi ,

where Yi is a vector field on Ui contained in E. Since X is an infinitesimal contact transformation if and only if

ix odyi +do tx y i =gi • yi

and since a 1-form is a multiple of yi if and only if it annihilates E, a necessary and sufficient condition for X to be an infinitesimal contact transformation is that

iz° ix° dY i+ lzo do ix yv=0 for all vectors Zek, xe ui . But this is equivalent to

tz o t y,odyi +Z(hi)=0 for ZeEx , xeLli .

Since dy i defines a non-degenerate bilinear form on Ex , this equation determines a unique vector Yi E Ex at each xe q. q. e. d.

7. Contact Structures 31

We shall now see, under the isomorphism a (M, y).—* II° (Al; L), how the Lie bracket looks like in H° (M; L). Let X and Y be two infinitesimal contact transformations and let s and t be the corresponding sections of the line bundle L, i. e.,

s=y(X)=yi (X) si and t = y ( Y)= y i (Y) si . Setting

f=yi (X) and g= y i (Y)

and defining [f, g] by

[f, g]=yi ([X, Y]) ,

we want to express [f, g] in terms of f and g. Since X is an intinitesimal contact transformation, we have Lx y l =hi • y i , where hi is a function defined on U. We shall first express hi in terms off. Let Si be the vector field on ui constructed in the proof of Theorem 7.1. Then X = f Si + X', where X' is a vector field on Ui with values in E. We have

hi • y i =Lx y i =do tx y i + ix ody g --=df +lx • o dy i .

Applying tsi to the both ends of the equalities, we obtain hi =Si (f). Thus,

Lx y i =df + tx ody i =Si (f) • y i ,

This formula allows us to recover X from f as follows. Taking ui suf-ficiently small we may assume that y i is expressed as

y1=x1dXm+1±x2dXm+2 +•••+e'dx 2 m+dx2 m+ 1

in terms of an admissible local coordinate system x', x 2 m+ 1 (see Appendix 1, Theorem 1). Then

dy i =dx1 dxm+ 1 + •••+dxm dx2 m and si=49/ax2m-1-1 .

By a direct calculation we obtain m a a a

k=1 axk k a xm+ k}± (f E X-Jk)

k=1 X = E tockf2m+i—fm+k) +f

where fh =aflaxh for h= 1, ...,2m+1.

From this expression we obtain rn

[fl g] = E Uk(gm+k—xkg2m+o—gkum+k—xkf2m+o) k=1

+fg2m+1- gf2m+1.

ax2m+1


The following result, proved first by Hatakeyama [1] in the compact case, is due to Boothby [3].

Theorem 7.2. Given a contact structure (Ui , yi ) on a manifold M, the group 91,(M, y) of contact transformations with contact support (in fact, already the subgroup generated by ( (M, y)) is k-fold transitive on M for every positive integer k.

Proof. In view of Lemma 1 in the proof of Theorem 6.1, it suffices to prove that the group generated by ac (M, y) is strongly locally transitive. Given a point p eM and a neighborhood U of p, let x1, x2 m+ 1 be a local coordinate system around p such that yi = x1 d xm + • • • + xm d x 2 m + dx 2 m + 1 (assuming that p e [It ). Let V be a small neighborhood of p with V Œu such that the above expression of yi in terms of the local coordinate system is valid. We shall construct a vector subspace b of (OM, y) of dimension 2m+ 1 ( =dim M) consisting of vector fields with support in V such that the mapping

Xeb —*(exp X)peM

gives a diffeomorphism of a neighborhood of 0 in b onto a neighborhood of p in M. Let B be the (2m + 1)-dimensional space of functions on V defined by

B= {f=a2m+1+ m E (ant+ k xk —ak xm+ k); (al , ... , a2m+i )eR 2 m+ 1 .

Let p be a function which is equal to 1 in a neighborhood of p and has support contained in V Set By ----{pf; feB). Then Bv is a (2m + 1)- dimensional space of functions on M with support in V Using the notation in the proof of Theorem 7.1 and multiplying each element of Bv by si , we consider By as a subspace of the space H° (M; L) of sections of the line bundle L. Let b be the subspace of a(M, y) corresponding to By under the isomorphism a(M, y)=1-1° (114; L) established in Theorem 7.1. Using the explicit formula above which reproduces a vector field X from f, we see that a function

f=a2m+i+ E (am, xic — ak xm+k) k.1

gives rise to

x= E tockf2.1—fm+k)—axk +fk aX m+k} + (f Exkfk) a X2n1+1 k=1

m E (ak Dx k +a

k=1 m+k axm+k) (a2m+1— Eakxm+k ax2m+1

k.1

8. Pseudogroup Structures, G-Structures and Filtered Lie Algebras 33

Consider the differential of the mapping p fe By (= b) —*(exp X) pE M at the origin O. It is a linear mapping of By = To (By) into Tp (M) which sends p fe By into X p e Tp (M). The formula above for X shows that

2m+1

Xp = E . Hence, the differential of By M at the origin is i=1 vx P

non-degenerate. The theorem now follows from the inverse function theorem. q. e. d.

We remark here that the use of the subbundle E and the quotient line bundle L of T(M) gives a simple proof of the following well known result, (cf. Gray [1]).

Theorem 7.3. A contact structure {El i , yi ) on M can be represented by a globally defined 1-form if and only if M is orientable.

Proof Since E is an orientable vector bundle, M is orientable e., the tangent bundle T(M) is orientable) if and only if the quotient bundle L is orientable. Since Lis a real line bundle, it is orientable if and only if it has a cross section without zeros. If s is such a section, then si = hi s, where hi is a function on Ui and si is the section of L over Ui characterized by yi (si)= 1 (see the proof of Theorem 7.1). Then the 1-form y= hi y i is well defined on M and satisfies the equation y(s)= 1. Conversely, if the contact structure (U, , y i ) can be represented by a globally defined 1-form y, then y =hi y i with a suitable function hi on U1 . The section s definéd by si = hi s is globally defined on M and satisfies y(s)= 1. q. e. d.

An example of a contact manifold is an odd-dimensional sphere S2171 + 1 . In en + with natural coordinate system z1 , ...,e+ 1 where Zk =x" + i yk , set y =E xl‘ dy k . Let S214 + 1 be the unit sphere centered at the origin in in + 1 . Then y induces a contact form on

Another example is the cotangent sphere bundle over any manifold M. Let M be an (m + 1)-dimensional manifold and T* (M) be the cotangent bundle. Let y be the 1-form on T* (M) constructed at the end of § 6. Choose any Riemannian metric on M and let 5* (M) be the unit sphere bundle consisting of covectors of length 1. Then y induces a contact form on S* (M).

For more information on contact structures, see Boothby-Wang [1 ], Gray [1], Libermann [2], Sasaki [1], Takizawa [1 ], Lichnerowicz [4].

8. Pseudogroup Structures, G-Structures and Filtered Lie Algebras

As we have seen in § 2, the concept of G-structure unifies a large number of interesting geometric structures. We shall now consider another unifying concept, namely, that of pseudogroup structure.


Let E be an n-dimensional manifold which will be taken as a model space. It is usually a Euclidean space. A pseudo group of transformations on E is a set T of local diffeomorphisms satisfying the following conditions:

(I) Each fer is a diffeomorphism of an open set (called the domain off) of E onto an open set (called the range off) of E.

(2) Let U

where each Ui is an open set of E. A diffeomor- i

phismf of U onto an open set of E belongs to i if and only if the restriction off to each Ui is in F.

(3) For every open set U of E, the identity transformation of U is in F.

(4) If f is in T, then f -1 is in F

(5) If fer is a diffeomorphism of U onto V and fief is a diffeomor-phism of U' onto V' and if Vn U' is nonempty, then the diffeom.orphism f' f off -1 (V n tf) onto f ' (V n U') is in T.

A pseudogroup T of transformations of E is said to be transitive if for every pair of points p and q of E, there exists an element f of T such that f (p). q.

Fix a transitive pseudogroup T of transformations of E. An atlas of a topological space M compatible with T (a T-atlas, for short) is a family of pairs (Ui , (p i), called charts, such that

(a) Each Ui is an open set of M and U U =M.

(b) Each cp i is a homeomorphism of Ui onto an open set of E.

(c) Whenever Ui U; is nonempty, the mapping Ti c) cpï' of T i (Ui nUi) onto (pi (U r U.i) is an element of E

A r-atlas of M is said to be maximal (or complete) if it is not contained in any other F-atlas of M. Every /1-atlas is contained in a unique maximal /1-atlas. A T-structure on M is a maximal F-atlas of M. It is customary to assume that M is a Hausdorff space. A T-manifold is a Hausdorff space M with a fixed maximal /1-structure. Every / 1-atlas of M, enlarged to a unique maximal r-atlas, defines a r-structure on M.

We shall give a few examples.

Example 8.1. Let E = R" and r be the set of all local diffeomorphisms of E. Then a r-manifold M is a usual (differentiable) manifold and a r-structure is a differentiable structure.

Example 8.2. Let E=R" with natural coordinate system x', x" and set co=dx 1 A • • • A dx". Let r be the set of all local diffeomorphisms f of E such that f* co= c co, where cf is a constant (which depends on f). Given a r-atlas ((q, (pi)) of M, set coi = cpt co. Then each coi is an n-form on Ui without zeros and coi = ci; coi on Ui 15 , where ci; is a (nonzero)


constant. Conversely, given a family of pairs (Ui , cot) such that Ui is an open cover of M and coi is an n-form on Ui without zeros satisfying coi =cii coi on Ui r' 15, we can recover a T-atlas of M and a unique T-structure on M.

Example 8.3. Let E and co be as in Example 8.2 and I' be the set of all local diffeomorphisms f of E such that f* co =co. Then the T-structures on M are in one-to-one correspondence with the volume elements of M in a natural manner. This structure was discussed in § 6.

Example 8.4. Let E=R 2 m with natural coordinate system x 1 , ... , x2 m and set co =dx1 A dxm+ 1 + • • • +dxm A dx2 m. Let T be the set of all local diffeomorphisms f of E such that f* co =cf • co, where cf is a constant. Then the T-structures are in a natural one-to-one correspondence with the conformal symplectic structures (see Example 2.8 and also § 6), provided m 2. We recall that if f is a local diffeomorphism such that f* co= co,. • co for some function cpf , then (1) f is necessarily a constant.

Example 8.5. Let E and co be as in Example 8.4 and I' be the set of all local diffeomorphisms f of E such that f* co = co. Then the T-structures are in a natural one-to-one correspondence with the symplectic structures discussed in Example 2.8 and in §6.

Example 8.6. Let E =R 2 m+ 1 with natural coordinate system x 1 , ..., x2 "' and set y = x1 d xm + 1 + • • • + xm d x 2 1" + dx2" 1 . Let r be the set of all local diffeomorphisms f of E such that f*y=cp f •y, where cpf is a function. Then the T-structures are in a natural one-to-one correspond-ence with the contact structures discussed in § 7.

Example 8.7. Let E=R" and G be a Lie subgroup of GL(n; R). Let T be the set of all local diffeomorphisms f of E such that at each point of the domain of f the Jacobian matrix Jf belongs to G. Then the F-structures are in a natural one-to-one correspondence with the integrable G-structures.

Example 8.8. Let E=R" and Gc GL(n; R) as in Example 8.7. Let I' be the set of all local diffeomorphismsf of E such that the Jacobian matrix .11 of f is constant and belongs to G. Then a T-structure is called a flat G-structure. If G = GL(n; R), a T-structure is known as an affine structure.

Example 8.9. Let E be a manifold on which a Lie group L is acting transitively and T be the set of all local diffeomorphisms f of E which can be obtained by localizing the elements of L. This example generalizes Example 8.8; taking E=R" and L = R" • G (=the subgroup of the group of affine transformations generated by the translations IV and a linear group G), we recover Example 8.8. If E=S" and L is the group of Möbius transformations (i. e., conformal transformations) of S", then a T-structure


is called a flat conformal structure. If E=pn (R) and L =PGL(n; R) ( = the projective general linear group), then a r-structure is called a flat projective structure.

In order to relate pseudogroup structures to G-structures, we shall consider transitive Lie pseudogroups. We shall not be concerned here with intransitive Lie pseudogroups; for various definitions of general Lie pseudogroups, see Ehresmann [3], Kuranishi [I], Libermann [3], Singer-Sternberg [I], Rodrigues [2].

Following Ehresmann we construct the bundle of r-frames over M, (cf. Kobayashi [8]). We fix a point of a model space E as the origin and denote it by O. If E= IV, we take the usual origin 0 of R. If V and V' are two neighborhoods of the origin 0 in E and if U and U' are two neighbor-hoods of a point x eM, two diffeomorphisms f: V.-1./ and f': U' are said to define the same r-frame at x if x= f (0). f 1 (0) and if f and f' have the same partial derivatives up to order r at 0 (in terms of local coordinate systems around 0 and x). The r-frame given by f is usually denoted byi*,; (f ). The set of r-frames of M, denoted by 7(M), is a principal bundle over M with natural projection 7r, n (fc; (f)) =PO), and with structure group G' (n) which will be now described. Let Gr (n) be the set of r-frames j;(g) at OE E; it forms a group with multiplication defined by

(g) 0 (g) = (g- 0

The group Gr (n) acts on 7(M) on the right by (nofo. (g) = A (f 0 g) for jr„ (f) E (M), (g)E Gr (n),

Clearly, 131 (M) is the bundle of linear frames over M with group (n)= GL(n ; R). Let I' be a transitive pseudogroup on E and fix a T-structure on M.

Let P r(M, I') be the subset of 7(M) consisting of r-frames A(f) such that f U —07 is a chart of the maximal T-atlas. Let G' (f) be the sub-group of Gr . (n) consisting of r-frames (g) at 0 e E such that g e F. If P r(M, F) is a submanifold of P r(M), then 7(M, f) is a subbundle of P r(M) with group Gr(F). Since E carries a natural T-structure itself, we can apply the construction of P r(M, I') to E to obtain P r(E, 1'). Since M and E are locally isomorphic as T-manifolds, P r(M, 1') .is a submanifold of Pr(M) if (and only if) 7(E, I') is a submanifold of Pr(E).

We say that a transitive pseudogroup I' on E is a Lie pseudogroup if P r(E, I') is a submanifold (and hence a subbundle) of (E) for every positive integer r and if there is a positive integer s with the property that a local diffeomorphism h of E is in T if the induced local automor-phism h* of P (E) leaves ?(E, I') invariant. (Roughly speaking, P (E, is a system of partial differential equations and the condition says that if h sends solutions into solutions, then h must be in f.) The smallest integer s satisfying the above condition is called the degree of T.


Let F be a transitive Lie pseudogroup of degree s on E and fix a F-structure on M. Then we have a subbundle (E, F) of P(E) and a subbundle F) of P (14). From Ps (E, F) we can recover the pseudogroup r by taking all local diffeomorphisms of E which leave (E, invariant, j. e., all local automorphisms of P5 (E, 1'). From Ps (M, I') we can reconstruct the T-structure of M by taking all local diffeomorphisms of M into E which maps Ps(M, I') into Ps(E, 11), i.e., all local isomorphisms of Ps(M, I') into Ps(E, F).

In order to unify the concept of G-structure and that of transitive Lie pseudogroup, we introduce the concept of G-structure of higher degree. Let G be a Lie subgroup of Gs (n). Then a subbundle P of P5 (M) with structure group G is called a G-structure of degree s on M. A G-structure in the sense of earlier sections is a G-structure of degree 1. Given G c Gs (n), let R" x G denote the natural (flat) G-structure of degree s on W. A diffeomorphism f of an open set U of M onto an open set of Rn is called an admissible local coordinate system if f induces an isomorphism of Pl u onto (It" x G)I f(u) . If every point of M has a neighborhood with admissible local coordinate system, then the G-structure P is said to be integrable. This generalizes the concept of integrability introduced in § 1.

We shall now reexamine Examples 8.1 through 8.9. The pseudo-groups F in these examples are all transitive Lie pseudogroups. Those in Examples 8.1, 8.3, 8.4 (for m 2), 8.5, 8.6, 8.7 are of degree 1 and, in each of these cases, G1 (F) is GL(n; R), SL(n; R), CSp(m; R), Sp(m; R), GL (n —1, 1; R) and G, respectively. The pseudogroups F in Examples 8.2 and 8.8 are of degree 2 . In case of Example 8.9, the degree s of F is the smallest integer with the following property: if g and g' are transforma-tions of E given by elements of L, jsc (g) = fcr 1 Igi‘ ) implies g =g'. The Gs(F) structure (11/1, I') is integrable in Examples 8.1, 8.2, 8.3, 8.4, 8.5, 8.7, 8.8 while it is never integrable in Example 8.6. In case of Example 8.9, P'(M, I') is sometimes integrable, e.g., when L is the group of Möbius transformations of Sn or when L is PGL(n; R) acting on F(R).

To each transitive Lie pseudogroup F we shall associate a transitive filtered Lie algebra. In general, we define a filtered Lie algebra to be a Lie algebra 1 (possibly of infinite dimension) with decreasing sequence of subalgebras 1=1_ 1 lo DI, D such that

(a)

(b)

(c)

[Ip Li] ip -Fq

dim I/Ç 1 < 00,

A filtered Lie algebra 1 is said to be transitive if

(d) 11, = {X eIp _ i ; [X, I] Ip _ 1 } for p-1.


As we have seen earlier, a graded Lie algebra is a Lie algebra g = such that

(a') [gp, gq] Œ gp-1-4 , (b') dim gp < oo.

A graded Lie algebra g is said to be transitive if

(d') [X, g _ *0 for every nonzero X E gp , p O.

Every (transitive) graded Lie algebra g may be considered as a (transitive) filtered Lie algebra in a natural manner, i.e., Ip = gp + g+1 + • • • . To each (transitive) filtered Lie algebra I we can associate a graded Lie algebra 9(1)— gp (I) by setting g(I)=I/I 1 and defining the bracket operation in a natural manner. Two non-isomorphic filtered Lie algebras may give rise to the same graded Lie algebra.

Given a transitive Lie pseudogroup I' acting on E, let a be the Lie algebra of germs -of vector fields X at 0 e E such that exp(t X) is in 11 for small values of t, Itl< 6. Let .97 denote the algebra of germs of functions defined around OeE and f be the maximal ideal of .97, e., the ideal consisting of germs of functions vanishing at O. We define a filtration a=a_ i mao mai p... by

ap ={Xea; X(.97)çfP+ 1 }.

In other words, X is in ap if and only if the components of X, expanded into Taylor series in terms of a local coordinate system around ()GE, have no terms of degree less than p. Then with this filtration the Lie algebra a satisfies conditions (a), (b) and (d). Set a.= n al, and define

I= a/a. and Ip ap/a.. Then the Lie algebra I with filtration I Io = I a ... is a transitive filtered Lie algebra. We note that a. consists of germs of vector fields X such that the Taylor series of the components of X (when expanded in terms of a local coordinate system around Oe E) are trivial.

This filtered Lie algebra I is useful in studying the automorphisms of a T-structure. But we shall not go into this question here.

For filtered and graded Lie algebras, see E. Cartan [5-7], Guil-lemin [2], Guillemin-Sternberg [1], Kac [1], Kobayashi-Nagano [3, 4], Ochiai [1], Singer-Sternberg [1], Shnider [1], Tanaka [5-7], Weis-feiler [1], Morimoto-Tanaka [1].

II. Isometries of Riemannian Manifolds

1. The Group of Isometrics of a Riemannian Manifold

The earliest and very general result on the group of isometries is perhaps the following theorem of van Danzig and van der Waerden [1] (see also Kobayashi-Nomizu [1, vol. 1; pp. 46-50] for a proof).

Theorem 1.1. Let M be a connected, locally compact metric space and 3(M) the group of isometries of M. For each point x of M, let 3.,(M) denote the isotropy subgroup of 3(M) at x. Then 3(M) is locally compact with respect to the compact-open topology and 3„(M) is compact for every x. If M is compact, then 3(M) is compact.

Eleven years later, in 1939, the following result was published by Myers and Steenrod Di

Theorem 1.2. The group 3(M) of isometries of a Riemannian manifold M is a Lie transformation group with respect to the compact-open topology. For each xe M, the isotropy subgroup 3„(M) is compact. If M is compact, 3(M) is also compact.

Before we begin the proof, we should perhaps point out that, a priori, there are two definitions of isometry for a Riemannian manifold. A dif-feomorphism f of M onto itself is called an isometry if it preserves the metric tensor. We can also call any one-to-one mapping of M onto itself which preserves the distance function defined by the Riemannian metric an isometry of M. According to Myers and Steenrod, these two definitions are equivalent (see Kobayashi-Nomizu [1, vol. 1; p. 169] for a proof). In this book, we adopt the first definition.

Let n =dim M. In the original proof of Myers and Steenrod, they took n + 1 points xo , xl , ..., x„ which are independent in a certain sense and proved that the mapping f e 3 (M) —* ( f (x 0), f (xi ), . . . , f (x„)) e Mn +1= Mx •••xM is one-to-one and has a closed submanifold of Mn+ 1 as its image. The have proved that the differentiable structure on 3(M) intro-duced by the injection 3(M)c Mn+ 1 makes 3(M) into a Lie transformation group. Theorem 1.2 may be also derived immediately from Theorem 3.3

40 II. Isometries of Riemannian Manifolds

(Bochner-Montgomery) of Chapter I and from Theorem 1.1 (van Danzig-van der Waerden). But we prefer to derive it from Theorem 3.2 of Chapter I as follows.

Proof of Theorem 1.2. Let L(M) be the bundle of linear frames over M; it is a principal bundle with group GL (n; R), n = dim M.

Lemma 1. Let 6= (01 5 ... 5 On) be the canonical form on L(M). For every transformation f of M, the induced automorphism f of L(M) leaves 0 invariant. Conversely, every fibre-preserving transformation of L(M) leaving 0 invariant is induced by a transformation of M.

Proof of Lemma 1. Let u eL(M) and X* e 7: (L(M)). We set X =n(X*)e T(M), where it: L(M)—* M is the projection and x =n (u). Then

9(X*)=14 -1 (X) and Ott X*)=/(u) -1 (fX),

where the frames u and f(u) are considered as linear mappings of Rn onto Tx (M) and Tf (x) (M), respectively. It follows from the definition of f that the following diagram is commutative:

Rn

Tx (M)— .,.' Tfix)(M). Hence, 1.4 -1(X)=Au) -1(f X), thus proving that 0 is invariant by f

Conversely, if F is a fibre-preserving transformation of L(M) leaving 0 invariant, let f be the transformation of the base M induced by F. If we set J= f' 0F, then J is also a fibre-preserving transformation of L(M) leaving 0 invariant and induces the identity transformation on the base M. Hence,

u-1(X) = 9 (X*) = 0(J X*)= .1 (u) - '(X) for X e 7: (L(M)).

This implies J (u) = u, that is, f(u) = F (u). Let a. ) = (coi)0 . 1 , ..., n be the connection form for an affine connection

of M. Then a transformation f of M is an affine transformation if f preserves co. From Lemma 1, we obtain

Lemma 2. Let 0 and co be the canonical form, and a connection form on L(M) respectively. If f is an affine transformation of M, then f preserves both 0 and co. Conversely, every fibre-preserving transformation of L(M) leaving both 0 and co invariant is induced by an affine transformation of M.

Lemmh 2 implies that the group 9I(M) of affine transformations of M is isomorphic to the group of bundle automorphisms of L(M) leaving both 0 and co invariant. On the other hand, the n + n' 1-forms 6 = (01 ) and co = (co. ) define an absolute parallelism, i.e., a (1)-structure, on

1. The Group of Isometries of a Riemannian Manifold 41

L(M). From Theorem 3.2 of Chapter I, it follows that the group 2I(M) of affine transformations may be considered as a closed submanifold of L(M). This result will be stated as Theorem 1.3 later.

Let M be a Riemannian manifold. Let 0(M) be the bundle of ortho-normal frames over M; it is a principal bundle with group 0(n). We denote by 0 and co the canonical form on 0(M) and the connection form for the Riemannian connection, respectively. A transformation f of M is an isometry if and only if the induced bundle automorphism f of L(M) leaves 0(M) invariant. We denote by f the restriction of f to 0(M). Since co is the unique torsionfree connection in 0(M), every bundle automorphism of 0(M) leaving the canonical form 0 invariant leaves.the connection form co invariant. Hence,

Lemma 3. Every isometry f of a Riemannian manifold M induces a bundle automorphism f of 0(M) leaving both the canonical form 0 and the connection form co invariant. Conversely, every fibre-preserving transformation of 0(M) leaving both 0 and co invariant is induced by an isometry of M.

On the other hand, w = (coi) is skew-symmetric, and the n(n +1) 1-forms 0 =(0i) and (co) define an absolute parallelism on 0(M). By Theorem 3.2 of Chapter I, the group 3(M) of isometries of M may be considered as a closed submanifold of 0(M). An imbedding 3 (M)c 0(M) is defined as follows. Choose an orthonormal frame uo e0(M). Then an imbedding is given by

fE 3 (M) f(uo) e 0 (M).

Under this imbedding, the isotropy subgroup of 3(M) at xo = i (u0) is the intersection of $ (M) and the fibre of 0(M) at xo in 0(M). Since each fibre of 0(M) is compact, the isotropy subgroup is also compact. If M is compact, so is the bundle space 0(M). Hence, its closed submanifold 3(M) is also compact. q. e.d.

We have proved not only Theorem 1.2 but also the following

Complement to Theorem 1.2. The differentiable structure of 3(M) is given by an imbedding of 3(M) in the bundle 0(M) of orthonormal frames as a closed submanifold as follows. If u0 is any orthonormal frame of M, then the mapping f e3(M)—)/(u 0)E0(M) defines an imbedding, where f is the bundle automorphism of 0(M) induced by f.

In the course of the proof, we have established also the following

Theorem 1.3. Let M be a manifold with an affine connection. Then the group 91(M) of affine transformations of M is a Lie transformation

42 IL Isometrics of Riemannian Manifolds

group. Its differentiable structure is given by an imbedding into the bundle L(M) of linear frames as a closed submanifold as follows. If u 0 is any linear frame of M, then mapping fE91(M) —> f( 140)E L(M) defines an imbedding.

Theorem 1.3 is originally due to Nomizu [1] and Hano-Morimoto [1]. The proof given here is due to Kobayashi. Theorem 3.2 of Chapter I was proved precisely to give a unified and geometric proof for the groups of isometrics, affine transformations, conformal transformations, pro-jective transformations and, more generally, automorphisms of a Cartan connections (see Kobayashi [1]). This proof gives also an upper bound for the dimension of any of these groups. The proof of Theorem 1.2 is a special case of the proof of Theorem 5.1 of Chapter I.

2. Infinitesimal Isometrics and Infinitesimal Affine Transformations

Although we are primarily interested in infinitesimal isometrics here, we consider also infinitesimal affine transformations at the same time. A vector field X on a manifold with an affine connection (resp. a Rie-mannian manifold) is called an infinitesimal affine transformation (resp. infinitesimal isometry or Killing vector field) if it generates a local 1-para-meter group of local affine transformations (resp. local isometrics).

For any vector field X on a manifold M with an affine connection whose covariant derivation is denoted by V, we define a derivation Ax by

Ax =Lx —Vx ,

where Lx denotes the Lie derivation with respect to X. Then (cf. Kobayashi-Nomizu [1, vol. 1; p. 235])

Proposition 2.1. For any vector fields X and Y on M, we have

Ax Y= — Vy X —T(X,Y),

where T is the torsion tensor field of the connection V.

Proof From Vx Y— Vy X —[X,Y]=T(X, Y) and Lx Y=[X, Y], we obtain Proposition 2.1. q. e. d.

In terms of a local coordinate system x', ... , xn, Proposition 2.1 means that Ax is the tensor field of type (1, 1) with components

_ vi v _E Tki i ec, . a

where X = i ax, .

For computational purpose, the following proposition is most useful.

Proposition 2.2. (1) A vector field X on a manifold M with an affine connection is an infinitesimal affine transformation if and only if

Vy (A x)=R(X, Y) for all vector fields Yon M,

2. Infinitesimal Isometries and Infinitesimal Affine Transformations 43

where R is the curvature tensor. In terms of a local coordinate system,

VI(Vi 1-FE

(2) A vector field X on a Riemannian manifold M is an infinitesimal isometry if and only if Ax is skew-symmetric, i. e.,

g(Ax Y, Z)+ g(Y, A x Z)=0 for all vector fields Y, Z on M,

where g is the metric tensor. In terms of a local coordinate system,

i +Vi &i =0.

Proof (1) We prove first

Lemma 1. A vector field X is an infinitesimal affine transformation if and only if

Lx 0 Vy Z - Vy 0 Lx Z =VEx, yj Z for all vector fields Y, Z on M.

Proof of Lemma 1. Assume that X is an infinitesimal affine transformation and let f be a local 1-parameter group of local transformations of M generated by X. Since J preserves the connection, we have

ft (Vy Vft y (f, Z) for all vector fields Y, Z on M.

From the definition of Lie differentiation, we obtain

Lx o Vy Z =11M -1

[Vy Z—f,(Vy Z)] 1-.0 t

1 1 [Vy Z - Vft y

t o t o t

=Vir.,xy Z+Vy 0Lx Z=V[x, y] Z+Vy 0Lx Z.

To prove the converse, assume the formula in Lemma. Fixing a point x of M, we set

=(f; (Vy Z))„ and W(t)=(Vf, y (ft Z))„.

For each t, both V(t) and W(t) are elements of T(M). As in the proof above, we obtain

dV(t) dt =f,((Lx o vy z)fc (x)),

dW(t) r7 t_y y ,71

=Jtk v[x, G Vy 0 L. x r 1 (x)) • dt


From our assumption, we obtain dV(t)Idt=dW(t)Idt. On the other hand, we have evidently V(0)= W(0). Hence, V(t)= W(t). This completes the proof of Lemma 1. •

From Lemma 1, it follows that X is an infinitesimal affine trans-formations if and only if

Lx O Vy Z Vy 0 L1 Z_(V1 0 Dy Z - Vy 0 DTI = Z - [ Dx Dy] Z or

Ax VyZ-Vy 0 A x Z=—R(X, Y)

for all vector fields Y, Z on M. But the left hand side is equal to - (Vy (A x)) Z. Hence, (Vy (A x)) Z =R(X, Y) Z.

(2) A vector field X is an infinitesimal isometry if and only if Lx g =O. Since g is parallel and, hence, Dv g=0, Lx g=0 is equivalent to

Ax g =O. Since Ax is a derivation of the algebra of tensor fields, we have

Ax(g(Y, Z))=(A x g) (Y, Z)+ g(A x Y, Z)+ g (Y, A x Z)

for all vector fields Y, Z. Since Ax maps every function into zero, the left hand side vanishes. Hence, Ax g =0 if and only if g(243‘ Y, Z) + g(Y, A x Z)= 0 for all Y, Z. q. e.d.

If X is an infinitesimal affine transformation of a Riemannian mani-fold M, then Proposition 2.2 implies

Vi Vi e+Ekiki 1`=0

and hence (by multiplying by g-ll and summing over j and I, we have)

vjvi t +E Ri; v=o.

On the other hand, applying the Laplacian A to the 1-form =E c dx 1 (see the formula for A in Appendix 3), we obtain

=E( j dx t

From these two systems of equations, we obtain

A =2>R, j c 1 dxt.

Theorem 2.3. Let M be a Riemannian manifold and X a vector field on M. Let c be the 1-form corresponding to X under the duality defined by the metric. If X is an infinitesimal isometry, it satisfies the following systems of differential equations:

=2E Rti dx 1 ;

R=0 (i.e., div X =0).

Conversely, if M is compact and X satisfies (1) and (2), then X is an infinitesimal isometry.

2. Infinitesimal Isometries and Infinitesimal Affine Transformations 45

Proof We have shown already that if X is an infinitesimal affine trans-formation, it satisfies (1). If X is an infinitesimal isometry, then V1 is skew symmetric in i and j (see Proposition 2.2) and its trace vanishes. This implies (2). To prove the converse, we may assume that M is ori-entable. (If M is not orientable, consider its orientable double covering.) We use the following integral formulas (S denoting the Ricci tensor):

$ S (X, X)— trace (Ax o tA x)— trace ((Ax +tAx) 2 )—(div X) 2 } dv =0

and {—(AX, X)+ S(X, X)+ trace (Ax 0 tA x)} dv=0.

The first formula is proved in Corollary to Theorem 1 in Appendix 2. The second formula is proved in Theorem 3 of Appendix 2; we note that

trace (Ax o tA x).E vi e o v./ j =(vx, vx). By adding these two integral formulas, we obtain

S (d X, X)+ 2 S (X, X)— trace ((A x -F lAx) 2 ) .—(div X) 2 } dv =0.

By our assumptions (1) and (2),

(AX, X)+2S(X, X)= 0 and div X=0.

(We note that AX is defined to be the vector field corresponding to the 1-form see Appendix 2.) Hence,

$ trace ((Ax + tAx)2). .

Since Ax +tA x is a symmetric tensor, trace((Ax -FlAx) 2) is the square of the length of A1 + 1A. It follows that Ax -FtAx =O. By Proposition 2.2, X is an infinitesimal isometry. q. e. d.

Theorem 2.3 and the following application is due to Yano [1 ].

Corollary 2.4. Let M be a compact Riemannian manifold. Then every infinitesimal affine transformation X is an infinitesimal isometry.

Proof We have shown already that X satisfies (1) of Theorem 2.3. In

ViViV+ER`jki k =0 we sum over i=j. Then

Vi (div X) =0

which means that div X is a constant function on M. On the other hand,

(div X)dv=0.

Hence, div X =0, showing that (2) of Theorem 2.3 is also satisfied. Now the corollary follows from Theorem 2.3. q. e.d.

46 II. Isometrics of Riemannian Manifolds

In general, an infinitesimal affine transformation or an infinitesimal isometry X generates only a local 1-parameter group of local affine transformations or local isometries. If it generates a global 1-parameter group of transformations, we call it a complete vector field. Thus, the group 91(M) of affine transformations (resp. the group 3(M) of iso-metries) of M has as its Lie algebra the set of complete infinitesimal affine transformations (resp. isometries) of M. We quote the following result (Kobayashi [4]).

Theorem 23. Let M be a manifold with a complete affine connection (resp. a complete Riemannian manifold). Then every infinitesimal affine transformation (resp. isometry) is complete.

We only sketch an outline of the proof. For detail, see Kobayashi-Nomizu [1, vol. 1; pp. 234 and 239].

Let L(M) be the bundle of linear frames over M. Let 61 =(0 and co.(ali) be the canonical form and the connection form on L(M). A vector field B on L(M) is called a standard horizontal vector field if tY(B)=constant and co(B)=O. Fix a point u0 in L(M). Then for each point u of L(M), there exist standard horizontal vector fields B1 , Bk and an element a of GL(n; R) such that

u=(bill 0 N2 0 • • • 0 bt u 0) a,

where each b it is the 1-parameter group of transformations exp tB i generated by B. Since the connection is complete, exp tBi is defined globally. (The geodesics on M are given as the projections of the orbits of exp tB with standard horizontal B.) Let X be the infinitesimal trans-formation of L(M) induced by X. It suffices to prove f;=exp tX is defined globally since ft =exp t X is the projection of ft . We set

j(u)=(b 1 0 1;1,22 0 • • • 0 b k (f7,(u0))) a

for the values of t for which .fi(u0) is defined. The fact that .4 (u) is defined independent of the choice of B1 , ...,Bk follows from the fact that B1 , Bk are invariant by X so that j; and exp tB commute.

On the question of extending a local (infinitesimal) isometry to a global one, see Kobayashi-Nomizu [1, vol. 1; pp. 252-256] and No-mizu [4].

3. Riemannian Manifolds with Large Group of Isometrics

We first consider the following extreme case.

Theorem 3.1. Let M be an n-dimensional Riemannian manifold. Then the group 3(M) of isometries is of dimension at most n(n +1). Ifdim 3(M).

n (n + 1), then M is isometric to one of the following spaces of constant

3. Riemannian Manifolds with Large Group of Isometrics 47

curvature:

(a) An n-dimensional Euclidean space W. (b) An n-dimensional sphere S".

(c) An n-dimensional projective space 11,(R). (d) An n-dimensional, simply connected hyperbolic space.

Proof. Since dim 0(M)=1 n(n +1) and 3(M) is a closed submanifold of 0(M), it follows that dim 3(M) n(n +1). Suppose dim 3(M)=

n(n +1). Since 3(M) is a closed submanifold of 0(M) and dim 3(M)= dim 0(M), it follows that either 3(M)=0(M) or 3(M) coincides with one of the connected components of 0(M). (Note that 0(M) has one or two connected component's according as M is non-orientable or ori-entable.) In any case, given a 2-dimensional subspace p of T(M) and a 2-dimensional subspace p' of T, (M), there is an isometry which sends p onto p'. This means that the sectional curvature determined by p coincides with the one determined by p'. This shows that M is a space of constant curvature. Since x and x' can be arbitrary points of M, we can conclude also that M is homogeneous and, hence, complete. If M is simply connected, then M must be one of (a), (b) and (d) (see, for example, Kobayashi-Nomizu [1, vol. 1; p. 265). If M is not simply connected, let be the universal covering manifold of M. Every infinitesimal isometry X of M induces an infinitesimal isometry ji of SI in a natural manner. Hence, n(n + 1)= dim 3(M) dim 3(k)

n(n +1). This implies that every infinitesimal isometry t of SI is induced by an infinitesimal isometry X of M. If we write M=A-4/T, where F is a discrete subgroup of 3(11-4), then I' must commute with the identity component .3° (k) of 3(g1). If gl=Rn, then 3° ( 14) is the group of proper motions and only the identity transformation commutes with 3'(J -1). If M = sr!, then 3° (/.4)= SO (n +1) and only +I e0(n +1) com-mutes with SO (n +1). If ft'l is a simply connected hyperbolic space, then 3(1I)=0(1, n) (=the Lorentz group of signature (+, —)) and 3° (M)= identity component of 0(1, n). In this case, the identity element is the only element of 3(I71) which commutes with 3° (g1). Theorem 3.1 follows now immediately, q. e. d.

The fact that dim 3(M)<1 n(n + 1) unless M has a constant curva-ture is classical (see, for example, Eisenhart [1]).

Theorem 3.2. Let M be an n-dimensional Riemannian manifold with n*4. Then the group 3(M) of isometries contains no closed subgroup of dimension r for n(n —1) + 1 <r < - n(n + 1).

Proof Let 6 be a closed subgroup of dimension r of 3(M) and let 6„ denote the isotropy subgroup of at x eM. Then 6„ is a closed subgroup of 0(n) by Theorem 1.1.


Lemma. For n44, 0(n) contains no proper closed subgroup of dimen-sion >1(n-1)(n-2) other than SO (n).

Proof of Lemma. Let 5 be a proper closed subgroup of 0(n). Con-sider 0(n) as a group acting on the homogeneous space 0(n)/5. Since 0(n) is compact, there is an invariant Riemannian metric on 0(n)/5. Since 0(n) is simple for n#4, 0(n) contains no non-discrete normal subgroup and hence acts on 0(n)/5 essentially effectively. This means that the dimension of 0(n) cannot exceed that of the group of isometries of 0(n)/b. If we set m=dim 0(n)/5, Theorem 2.1 implies •

4n(n-1)=dim0(n)..qm(m+1), that is,

nm+1. This implies

dim 5=dim 0(n) —m_In(n-1)—(n-1)=-1(n-1)(n-2),

thus completing the proof of Lemma. Suppose r >in(n —1)+1. Then

dim S„ dim % — dim M>In(n-1)+1—n=1-(n (n — 2)+ 1

From Lemma, it follows that Sx =0(n) or Sx =S0(n). We shall show that is transitive on M. If x and y are two points of M which can be joined by a geodesic, let z be the midpoint of this geodesic segment and let Z be the vector tangent to the geodesic at z. Letfbe a transforma-tion belonging to Sz such that f* (Z)= —Z; such an isometry exists since Sz =--0(n) or Sz = SO (n). Clearly, f (x)= y and f(y)=x. If x and y are arbitrary points of M, we join them by a finite number of geodesic segments and apply the construction above to each segment. In this way, we see that there is an element of (6 which sends x into y. Since (6 is transitive on M, we have

r=dimS=dimM+dim Sx =n+dim0(n)=-1-n(n+1). q.e.d.

Theorem 3.2 is due to H.C.Wang [1]. Lemma used above is due to Montgomery and Samelson [1].

In view of Theorem 3.2, it is natural to ask which Riemannian manifolds of dimension n admits a group of isometries of dimension

n (n —1)+ 1. Let M be an n-dimensional Riemannian manifold with n#4. Let E•

be a closed subgroup of dimension In(n —1)+1 of 3(M). Let 15x be the isotropy subgroup of at x e M. We shall show that is transitive on M. Assume that it is not. Then, for every xeM, the orbit of e• through

3. Riemannian Manifolds with Large Group of Isometries 49

x is of dimension less than n. Hence,

dim 05„.clim —(n-1)=1n(n-1)+1—(n-1)=1(n-1)(n —2)+1.

By Lemma for Theorem 2.2, either 6„ =0(n) or 6„=S0(n). Then, as in the proof of Theorem 3.2, we see that t is transitive on M. Thus, M is a homogeneous Riemannian manifold 6/5, where 5 is a compact group of dimension 1(n —1)(n —2) (=dim 6—n).

Lemma 1. Let 5 be a connected closed subgroup of SO(n). If n*4, then 5 is isomorphic to either SO(n —1) or the universal covering group of SO (n —1). If n* 4, 7, then 5 is imbedded in SO (n) as a subgroup leaving a 1-dimensional subspace of Rn invariant. If n =7, then either 5 = SO (n —1) leaving a 1-dimensional subspace of Rn invariant or 5 = Spin(7) with the spin representation.

Proof of Lemma I. We shall prove only the first statement and indi-cate a proof for the remainder of Lemma 1. With respect to an invariant Riemannian metric on the homogeneous space SO (n)/5, the group SO (n) acts as a group of isometries. Since SO(n) is simple for n*4, its action on SO (n)/5 is essentially effective. Since dim SO (n)/$3 =n —1 and dim SO(n)=1n(n —1), Theorem 2.1 implies that SO(n) is a maximal dimensional isometry group acting on SO(n)/b and that SO(n)/5 is either a sphere or a real projective space. Under the linear isotropy representation, 5 is mapped onto SO(n — 1). Hence, 5= 50(n-1) or 5= Spin (n —1). The second and third statements tell us how SO (n —1) or Spin(n —1) can be imbedded into SO(n). The second statement is proved in Montgomery-Samelson [1] by a topological method. We indicate an algebraic proof. First, assume that the action of 5 on Rn is reducible with a p-dimensional invariant subspace W. Then it leaves an (n —p)-dimensional orthogonal complement Rn-13 invariant. Hence,

dim b_dim 0(p)+dim 0(n —p)=1p(p —1)+1(n —p)(n—p-1).

This implies that p =1 or p=n —1. Next, assume that 5 acts irreducibly on W. Then 5 is absolutely irreducible; otherwise, 5 would be a sub-group of U (n/2) of dimension n'. Now the problem is reduced to that of determining the irreducible representations of degree n of o(n —1; C). But this can be easily accomplished by means of the theory of represen-tations of semi-simple Lie algebras. q. e. d.

Lemma 2. Let 0 be a connected Lie group of dimension in (n —1)+1 and 5 a connected compact subgroup of 0 of dimension 4 (n —1) (n — 2) such that its linear isotropy representation at a point of M = 0/5 leaves a 1-dimensional subspace of the tangent space invariant. Let

g =1) +m' + m" (vector space direct sum)

50 IL Isometries of Riemannian Manifolds

be an (ad 6)-invariant decomposition of the Lie algebra g, where m' and m" are subspaces of dimension 1 and n-1, respectively. Then there are the following three possibilities, provided n >4:

(1) [1), nti =0, [m', "] =0, [m", m"] =0;

(2) [1), nil =0 > Ent% nil = 0 > [nt", re] = 1);

(3) [b, nti =0, [m', m"] =m", [m", mil =0 .

and [X,Y]=c Y for X em', Ye m", where c is a constant which depends only on X.

Proof of Lemma 2. Since the linear isotropy representation of 6 is of the form

/1 0 \ k0 SO(n-1)/ '

6 leaves m' elementwise fixed, so that [I), nti =0. We shall show that

either [m', m"] =0 or [m', m"] =m".

Fix a non-zero element X of m'. Since the kernel of the linear mapping Yem" —* [X, 1 ] e[m', m"] is invariant by ad 6, it must be either 0 so that dim [m', m"] = dim m" =n-1 or the whole space m" so that [m', m"]=0. We assume dim [m', nt"] =n-1. Since I), nt' and nt" have mutually distinct dimensions so that the irreducible representations of 6 on 1), m' and nt" are mutually inequivalent, it follows that the (n —1)- dimensional subspace [ni', mil of g =1)-F m' +nt" invariant by 6 must coincide with m". (Here, we used the assumption n >4.) This proves our assertion.

We shall show that if [m', nt"]=nt", then

[X,Y]=cY for X em' and Ye m",

where c is a constant which depends only on X and not on Y Since X em' is invariant by 6, the linear isomorphism Ye ni" -+ [X, Ile [m', nt"]=m" commutes with the action of 6 on nt". But 6 acting on nt" is nothing but SO(n —1). Hence, this linear isomorphism is a scalar multiple of the identity transformation.

We shall show that

either [m", ne] =0 or [ni", nt"] =1) .

Choose a unit vector Xi em' and an orthonormal basis X2 , ..., X„ for nt". Define the constants C . i c (i, j, k =1, ... , n), by

[Xi , XI ]=E C iik X1 mod 1) (with Ciii c = — CO .

3. Riemannian Manifolds with Large Group of Isometrics 51.

We have to prove C iik --.0 for 1 -.I .._n and 2 ..j, ic.n. Fix i, j, k. Choose an integer 1, 2.1-n, such that /4 i, j, k. Since n>4, this is possible. Let A be the linear transformation of m' + nt" defined by

A(Xj). — Xi , A(X 1)= —X,, A(X)=X p for p*j,I.

Since A belongs to SO(n —1), it is induced by an element a of $3. From

(ad a)([Xi ,Xd)= [(ad a) Xi , (ad a) XJ,

we obtain the desired relation ciik =0 by comparing the coefficients of Xi on both sides. Thus, [m", In"] ct). Since [m", ITV] is an ideal of b and since b is simple for n >4, we have either [in", nt"] =0 or [m", nt"] = b.

Finally, we prove that

[nt", "] =t) implies [nt', in"] =0.

Let X e m' and Y, ZEM" be nonzero elements such that [Y, 4 *0. Then

[X, [Y, Z]]=[[X, Y], Z] +[Y, [X, Z]]=[c: Y, Z]+[Y, c:Z] =2c[Y, Z].

On the other hand, from [in', 1)] =0, we obtain [X, [ Y, Z]]=0. Hence, c =0. This completes the proof of Lemma 2.

We shall now consider the case where $3 = Spin (7).

Lemma 3. Let 15 be a connected Lie group of dimension 29 ( =4.- 8 (8 — 1) + 1) and $3 = Spin (7) such that the linear isotropy representa-tion at a point of M =6/6 is the spin representation. Let

g = b + In (vector space direct sum)

be an (ad 5)-invariant decomposition of the Lie algebra g. 'Then

[In, m] =0 .

Proof of Lemma 3. Since dim b = 21+8 =dim in, the representations of $ on b and in are mutually inequivalent. On the other hand, [nt, In] is an 6-invariant subspace of g and has dimension 28 ( =48(8 —1)) since dim nt =8. Hence, we have [nt, in] =0, [m, In] = nt or [nt, nt] = b.

Assume [in, tn] = in. Let r be the radical of g. It is invariant by b. On the other hand, since b is simple, it follows that dim r dim g — dim b = 8. Hence, r =nt or r =0. Since [m, In] =m, in cannot be solvable. Hence, r =0, i.e., g is semi-simple. Since In is an ideal of g, there is a com-plementary ideal b'. From dim tr =dim t) and from the fact that the representations of $3 on b and In are inequivalent, it follows that the $3-invariant subspace b' must coinéide with b. Hence, [I), in] =0. This contradicts the assumption that the linear isotropy representation of 6 is irreducible so that [b, nt] =nt. We have thus excluded the case [in, m] =m.


Assume [m, m] = I). Let a be an ideal of g. Since the representations of 6 on and nt are inequivalent, an 5-invariant subspace of g must be g, h, nt or O. Hence, a must be g, h, nt or OE But neither h nor nt can be an ideal of g. Hence, either a =g or a =O. This shows that g is simple. But there is no simple Lie algebra of dimension 29. We have thus excluded the case [nt, nt] = h. This completes the proof of Lemma 3.

In Lemma 2 and 3, the n-dimensional Riemannian manifolds M such that 3(M) contains a closed subgroup of dimension In(n-1)+ 1 have been locally determined for n >4. We shall now consider the global classification.

Consider first the case (1) in Lemma 2. Clearly, M is locally sym-metric and flat. If M is simply connected, then M =1In =R X IVI-1 and (6 is the direct product of the group of translations on R and the group of proper motions of R". To find a non-simply connected M, we have to look for a discrete subgroup /1 of the group of motions of Rn which commutes with the above 0 elementwise. It is easy to verify that must be generated by a translation of R. In other words, if M is not simply connected, then M = SI. X where 51 denotes a circle.

Consider the case (2) in Lemma 2. Clearly, M is locally symmetric and reducible. If M is simply connected, then M =R x M", where M" must be a space of constant curvature by Theorem 3.1. Since [nt", nt"] = M" is non-flat When M is simply connected, 0 is the direct product of the group of translations of R and the largest connected group of isometries of M". This second factor is SO(n) or the identity component of the Lorentz group 0(1, n-1) according as the curvature is positive or negative. It is easy to see that a discrete subgroup of Z(M) commut-ing with 0 is generated by a translation of R and by — /e 0(n) if the curvature is positive and by a translation of R if the curvature is nega-tive. In other words, if M is not simply connected, then M = x 5n -1 , M =R x i_ 1 (R), M = x pn _ (R) or M is a product of R with an (n —1)-dimensional hyperbolic space.

We consider now the case (3) of Lemma 2. We shall show that g is a sul;algebra of the Lie algebra o(1, n) of the Lorentz group 0(1, n). Let

o(1,n)=f -Fp be the Cartan decomposition, j. e., is the Lie algebra of a maximal compact subgroup St of 0(1, n) and p is the orthogonal complement of

with respect to the Killing form. The symmetric space associated with this Cartan decomposition is a hyperbolic space. Choose a St-invariant inner product in p, i. e., an invariant Riemannian metric on the associated symmetric space, in such a way that the sectional curvature is —1. Then, if X, Y and Z are three vectors in p, then

R(Y, Z) X = [X, [ Y, Z]] = (X, Y) Z —(X, Z) Y,

3. Riemannian Manifolds with Large Group of Isometrics 53

where R is the curvature and ( , ) is the inner product in p. Choose a unit vector X in p and let be the 1-dimensional subspace of p spanned by X. Let p" be the orthogonal complement of nt' in p. Define a sub-space nt" of o(1, n) by

tn" = (Z+ [X, Z]; Zep").

Let t be the subalgebra of defined by

It is not difficult to see that the subalgebra 1)+ne +nt" of o(1, n) thus defined is isomorphic to the Lie algebra g in (3) of Lemma 2. In veri-fying this assertion, one should choose X in Lemma 2 in such a way that the constant c is equal to 1 and also make use of the relation [X, [ Y, Z]]=(X, Y) Z— (X, Z) Y above.

The correspondence

ep" Z + [X, Z] em"

defines a linear isomorphism between p =n11 + p" and m =m1 +m". On the one hand, p is identified with the tangent space of the symmetric space 0(1, n)/$t at the origin and has a natural inner product ( , ) corresponding to the invariant Riemannian metric of curvature —1. On the other hand, nt can be identified with the tangent space of the homo-geneous space M= E0/6 at the origin and has an inner product ( , which corresponds to the given Riemannian metric of M. Under the isomorphism between p and nt, these two inner products ( , ) and ( , )' may not correspond to each other. But they are none the less closely related to each other. Since ( ,

)' is invariant by SO (n —1) (which is the

subgroup of SO (n) leaving the 1-dimensional subspace nt' of m invariant), there exist positive constants a and b such that

(X, X)1 = a(X, X) for X e m'

(Z + [X, Z + [x, =b(Z, Z) for Ze p".

In other words, if we define a new inner product ( , )" on m by

(X, X)" =71 (X, X)' for X E nt'

(Y, Y)" =(Y, Y)1 for Ye ni",

then the corresponding new invariant Riemannian metric on M= eqb has constant negative curvature. We claim that M is simply connected. Although this may be proved in the same way as a similar assertion in the case (2), it is an immediate consequence of the general result that a


homogeneous Riemannian manifold with negative curvature must be simply connected (see Kobayashi-Nomizu [I, vol. 2; p. 105]).

Consider now the case of Lemma 3. Then M is clearly locally sym-metric and flat. If M is simply connected, then M =R8 and 0 is a semi-direct product of the group of translations and Spin (7) in an obvious manner. (The group of proper motions of R8 is a semi-direct product of the group of translations and SO(8). Since Spin(7) is a subgroup of SO(8), 0 is a subgroup of the group of proper motions of R 8 in a natural manner.) From the fact that Spin (7) acting on R8 is absolutely irreducible, it follows that the identity transformation is the only motion of R8 which commutes with 6. Hence, M must be simply connected.

What we have proved may be summarized as follows.

Theorem 3.3. Let M be an n-dimensional Riemannian manifold with n >4 such that its group 3 (M) of isometries contains a closed connected subgroup of dimension 5 n(n-1)+1. Then M must be one of the fol-lowing:

(1) M =R x V, where V is a complete simply connected space of con-stant curvature and 0=R x 3° ( 7);

(2) M =51 x V, where V is a complete simply connected space of con-stant curvature and 0=51 x (V);

(3) M=R x 11_ i (R) and 0 =R x 3° (11_ i (R)); (4) M=S' x g_ 1 (R) and 0=S' x 3°01_, at»; (5) M is a simply connected homogeneous Riemannian manifold 0/5

with a 0-invariant unit vector field X and admits a 0-invariant Rieman-nian metric of constant negative curvature (which agrees with the orig-inally given metric on the tangent vectors perpendicular to X).

If n=8, then the following additional case is possible:

(6) M =R 8 and (6=R 8 Spin (7) (semi-direct product), where R 8 denotes the translation group on R 8 and Spin(7) is considered as a sub-group of the rotation group S0 (8).

Remark. A precise description of 15 in (5) as a subgroup of the Lorentz group 0(1, n) is given in the discussion preceding the theorem. Its Lie algebra is described in (3) of Lemma 2.

A local version of Theorem 3.3 is essentially due to Yano [2] although he excluded the case n=8 from consideration. The global version given here is essentially due to Kuiper [3] and Obata

As we have shown above (see the paragraph preceding Lemma 1), if a four-dimensional group of isometries acts on a 3-dimensional Rie-mannian manifold, the action is transitive. E. Cartan ([8; pp. 293-306]) has classified all such groups together with their actions.

4. Riemannian Manifolds with Little Isometries 55

In the 4-dimensional case, difficulties arise from the fact that SO(4) is not simple. The 4-dimensional homogeneous Riemannian manifolds have been studied by Egorov [10] and Ishihara [1].

An extensive work on the dimension of the automorphism group of a Riemannian or affinely connected manifold has been done by Egorov [1, 15]. For a survey of the results on this subject obtained before 1956, see Yano [3]. Mann [1] has shown that if M is an n-dimensional Rie-mannian manifold, then 3(M) contains no compact subgroups of dimension r for

1(n — k)(n—k+1)+1-k(k+1)<r<+(n—k+1)(n—k+ 2)

for k=1, 2, ...,n,

except for n =4, 6, 10; his result generalizes Theorem 3.2 when M is compact. See also Janich [1]. For the determination of 15 c3(M) of dimension r. 1(n —1) (n — 2)+ 2 (compact or noncompact), see Koba-yashi-Nagano [5] and Wakakuwa [1].

Fix a manifold M of dimension n and, for each Riemannian metric ds2 on M, let 3(M, ds 2 ) denote the group of isometries of M with respect to ds 2 . Following W. Y. Hsiang [1-3], we define the degree of symmetry of M to be the maximum of dim .3(M, ds 2) for all possible Riemannian metrics ds 2 of M. It is a non-negative integer not exceeding 1 n (n +1). Define also the degree of compact symmetry of M to be the maximum dimension of all possible compact Lie groups acting on M. If % is a compact Lie group acting on M, there is a 15-invariant Riemannian metric on M. Hence, the degree of compact symmetry of M does not exceed the degree of symmetry of M. If M is compact, 3(M, ds 2) is compact and hence the two degrees coincide. The results in this section may be interpreted in terms of these degrees. See also Ku-Mann-Sicks [1] on the degree of symmetry.

4. Riemannian Manifolds with Little Isometries

In the preceding section, we considered Riemannian manifolds which admit many isometries. We consider here those which admit hardly any isometries. We begin with the following theorem of Bochner [1].

Theorem 4.1. Let M be a compact Riemannian manifold with negative Ricci tensor. Then the group 3 (M) of isometries is finite.

Proof Since 3(M) is compact by Theorem 1.2, it suffices to show that dim 3(M)=0, i.e., M admits no infinitesimal isometries. Let X be a vector field on M. We make use of the tensor field Ax of type (1, 1) defined by Ax =Lx —Vx as a derivation in § 2. Since the Riemannian


connection is torsionfree, we have Ax Y= —V y X for all vector fields Y, i.e., Ax = —VX (see Proposition 2.1). We proved (in Proposition 2.2) that X is an infinitesimal affine transformation if and only if

Vy (Ax)=R(X, Y) for all vector fields Y.

Lemma 1. Let x be a point of M and VI , ..., v„ an orthonormal basis for the tangent space Tx (M). If X is an infinitesimal affine transformation of M, then n

g (

E (Dv, Ax) Vi , X) = S (X , X), i.1

where S denotes the Ricci tensor and g the metric tensor.

Proof of Lemma 1. Since Vv, Ax =R(X, VA we have

g (

n n

E (Dv, Ax) Vi , X) = E g (R(X, Ili) Vi , X)=S(X, X),

thus proving Lemma 1. If f is a function on M, its second covariant derivative V 2f may be

considered as a covariant symmetric tensor field of degree 2 and defines a symmetric bilinear forms on each tangent space. If VI , ..., v. is an orthonormal basis for Tx (M) as above, then the Laplacian (A f)„ of f at x is given by

(Af)x = E v 2 f (vi, I'). il..'

Lemma 2. Let X be an infinitesimal affine transformation of M and f the function on M defined by f=i-g(X, X). Then

(1) V 2 f (V, V) = g (Vv X, W X) — g (R (X , V) V, X) for ye T(M); n

(2) (A f)= E g(Vvi X, Vv, X)— S(X, X), i=1

where VI , ..., V„ is an orthonormal basis for T(M).

Proof of Lemma 2. We extend each V to a vector field in a neighbor-hood of x in such a way that Dv V=0 at x. This can be accomplished by parallel displacing V along the geodesic exp(t V) for small values t and then extending it around the geodesic segment. Then

V 2 f (V, V)x =(V(. V f)) x =g(Vv X, Vir X)„+g(Vv Vif X, X)x

= g (Vv X, Vv X)x — g(Vv(Ax VI X)x =g(Vv X, Vv X)„ — g((Vv Ax) V, X)x — g(Ax(Vv V), X)x

=g(Vv X, Vv X)„ —g(R(X, V) V, X)x,

n

4. Riemannian Manifolds with Little Isometrics 57

thus proving (1). Applying (1) to V. V, and summing the resulting equalities with respect to i, we obtain (2).

To complete the proof of Theorem 4.1, we apply Green's theorem

f dv=0 Af

to the function f=lg (X, X). On the other hand, if X is an infinitesimal affine transformation, then A f >0 by (2) of Lemma 2 unless X is identi-cally zero. In applying Green's theorem, we have to assume that M is orientable. If M is not orientable, we have only to consider an orientable double covering of M. q. e. d.

From the proof above we obtain also the following result.

Corollary 4.2. If M is a compact Riemannian manifold with negative semi-definite Ricci tensor, then every infinitesimal isometry of M is a parallel vector field.

Theorem 4.1 may be generalized to a non-compact manifold to some extent.

Theorem 4.3. Let M be a Riemannian manifold with negative definite Ricci tensor. If the length of an infinitesimal isometry X attains a local maximum at some point of M, then X vanishes identically on M.

Proof Suppose f=4g(X, X) attains a local maximum at x. Then (Af)0. On the other hand, (2) of Lemma 2 above implies that d f >0 wherever X *0. Hence, X must vanish at x. Since f attains a local maxi-mum at x, this means that X vanishes in a neighborhood of x. By apply-ing "Complement to Theorem 1.2" to the local 1-parameter group generated by X, we see that X vanishes everywhere on M. q. e. d.

The following result is due to Frankel [3]. With a stronger assump-tion than in Theorem 4.1, we can prove a little more.

Theorem 4.4. Let M be a compact Riemannian manifold with non-positive sectional curvature and with negative definite Ricci tensor. An isometry f of M which is homotopic to the identity transformation must be the identity transformation..

Proof Let h„ :).t 1, be a homotopy such that ho is the identity transformation and h1 =f Let fi be the universal covering space of M with covering projection it : Let h, be the unique lift of h, such that It o is the identity transformation of M. (By a lift of h„ we mean h,: Ït( —la such that it 0 hi =h, o it.) We set f= Ti l . Then f is a lift off and, hence, is an isometry of M. Since each transformation h, normalizes the group of deck-transformations which is a discrete group, the 1-para-meter family h, of transformations must commute with the deck-trans-


formations elementwise. Since the deck-transformations are all isometries of JÇI, it follows that, for every p ek, the distance d(, f()) between p and f() depends only on it(). Hence, since M is compact, d(, f()) attains an absolute, hence relative maximum at some point po of M. We wish to calculate the Hessian and then the Laplacian of this function d (AM» at po

Let c be the geodesic from po to f p 0 ) ; since k" ( is simply connected and complete with non-positive sectional curvature, this geodesic exists and is unique (see, for example, Kobayashi-Nomizu [1; vol. 2; p. 102]). For each unit tangent vector X0 at po perpendicular to the geodesic c, we define a Jacobi field X along c extending X0 as follows. Consider the geodesic exp (s X0), IsI <, through po . For each fixed s, let cs be the geodesic joining the point exp (s X0) to the point f (exp (s X0)). Then cs , Is' <a, is a 1-parameter family variation of the geodesic segment C. Let X be the infinitesimal variation, i. e., the Jacobi field, defined by the variation cs (cf. Kobayashi-Nomizu [1; vol. 2; p. 63]). Denote by T the unit vector field tangent to the geodesic c, i. e., the velocity vector field of C. The formula for the first variation of arc-length is given by (see, for example, Kobayashi-Nomizu [1, vol. 2; p. 80])

dL(X) = g (X, nf(po) - g (X, .

Since the length L(s) of cs is the distance between exp(s X0) and f(exp(s X0)) and c is the longest curve in the family cs by hypothesis, we have dL(X)=0. Since g(X, npc, =0, it follows that g(X, T) .7(0) =0. Since the Jacobi field X is perpendicular to Tat two points, it is perpen-dicular to T everywhere along c. The second variation /(X, X)= (d 2 Lid s 2 )s ,_ 0 is given by (cf. Kobayashi-Nomizu [1, vol. 2; p. 81])

a

I(X, X)-= (g (X', X') — g(R (X, T) T, X)) dt (where X' =VT X), 0

where t denotes the arc-length parameter for c which is parametrized by 0

Let VI , 1/n _ i be tangent vectors at po such that T, V1 , ..., Vn _ i form an orthonormal basis of Tpo (ft). To each 14, we apply the construction above to obtain a Jacobi field, denoted also by V , along c. Then

n-1 a n-1

i(vi , -S E ORO/ , T, V i)dt. i.1 o

By our assumption on the sectional curvature the integrand is non-negative along c. If we denote by S the Ricci tensor as before, then the integrand coincides with S(T, T) since T, V„, is an ortho-

normal basis for Tplo (ft). Hence, E I( Vi , 14) >0 if c has positive length. i=1

5. Fixed Points of Isometrics 59

Thus, if c has positive length, then / (Ili , ili)>0 for some Iii , which contra-dicts the fact that c is the longest of the curves. Hence, c must have zero length, i. e., f (p 0)= p 0 . Since the distance (,f (p)) attains a relative maximum (actually an absolute maximum) at po , it follows that f (p)= il• in a neighborhood of po and hence everywhere on M. q. e. d.

We have assumed that in the homotopy ht each k is a diffeomorphism of M onto itself. The full strength of compactness of M was not used in the proof; the theorem still holds if the function F(p)= distance (3, f()), where p = it (P), attains a relative maximum at some point of M. The function F on M plays the same role as the length of an infinitesimal isometry X in the proof of Theorem 4.1 or 4.3. This function has been systematically exploited by Ozols [1 ].

For non-differentiable versions of some of the results in this section, see Busemann [2 ].

The following result of Atiyah-Hirzebruch [1] implies that a com-pact manifold M with nonzero Â-genus cannot admit a Riemannian metric for which dim 3 (M)> 0, i. e., its degree of symmetry is zero.

Theorem 4.5. If a circle group acts differentiably on a compact mani-fold M, then the Â-genus of M vanishes.

5. Fixed Points of Isometrics

The following elementary result shows that the fixed point set of a family of isometries is a nice differential geometric object.

Theorem 5.1. Let M be a Riemannian manifold and 45 any set of iso-metries of M. Let F be the set of points of M which are left fixed by all elements of 45. Then each connected component of F is a closed totally geodesic submanifold of M.

Proof Assuming that F is non-empty, let x be a point of F. Let V be the subspace of Tx (M) consisting of vectors which are left fixed by all elements of 15. Let U* be a neighborhood of the origin in Tx (M) such that the exponential mapping expx : U* .—lti is an injective diffeomorphism. Let U =expx ( U*). We may further assume that U is a convex neighbor-hood. Then it is easy to see that U n F =expx (U* n 1/). This shows that a neighborhood U n F of x in F is a submanifold exp x ( U* n 1/). Hence F consists of submanifolds of M. It is clear that F is closed. If two points of F are sufficiently close so that they can be joined by a unique mini-mizing geodesic, then every point of this geodesic must be fixed by 15. Hence, each component of F is totally geodesic. q.e.d.

Remark. More generally, if M is a manifold with an affine connection and 15 is a set of affine transformations of M, then the set F of points left


fixed by 0 is a disjoint union of closed, auto-parallel submanifolds (see Kobayashi-Nomizu [1, vol. 2; p. 61]).

The following result shows that the number of connected components in F is limited.

Corollary 5.2. In Theorem 5.1, assume that M is complete. Let p and q be points belonging to different components of F. Then q is a cut point of p. If sY• is a connected Lie group of isometries, then q is a conjugate point of p.

Proof If q is not a cut point, then by definition there is a unique minimizing geodesic, say c, from p to q. If f is any element of 0, then f (c) is also geodesic from p to q with the same arc-length as c. Hence c = f (c) and every point of c is left fixed by f Since f is an arbitrary point of 0, this shows that c is contained in F and, hence, p and q are in the same connected component of F.

Let 0 be a connected Lie group with positive dimension. Assume that q is not conjugate to p. Let c be a geodesic from p to q. Every infini-tesimal isometry X defines a Jacobi field along c; the one-parameter group generated by X defines a variation of c in a natural manner. If X belongs to the Lie algebra of 0, then X vanishes at p and q. Since q is not conjugate to p, X must vanish at every point of c. Thus, the 1-para-meter subgroup of 0 generated by X leaves c fixed pointwise. Since 0 is connected and is generated by these 1-parameter subgroups, leaves c fixed. This shows that p and q can be joined by a curve c contained in F. q. e. d.

Theorem 5.1 can be strengthened if 05 is a 1-parameter group. Indeed, we have the following result (Kobayashi [5]).

Theorem 5.3. Let M be a Riemannian manifold and X an infinitesimal isometry of M. Denote by Zero (X) the set of points of M where X vanishes. Let Zero (X). H N. be the decomposition of Zero (X) into its connected v -

components. Then:

(1) Each Ni is a closed totally geodesic submanifold of even codimension.

(2) Considered as afield of linear endomorphisms of T(M), the covariant derivative V X (= — A x) annihilates the tangent bundle T(N) of each Ni and induces a ( skew-symmetric) automorphism of the normal bundle V (N1) of each Ni . Restricted to each N1 vx is parallel, Vv (V X)=0 for every ye T(1Vi).

(3) The normal bundle T I (Ni) of Ni can be made into a complex vector bundle.

(4) If M is orientable, then each Ni is orientable.

0

0 ak —ak 0

5. Fixed Points of Isometries 61

Proof Applying the proof of Theorem 5.1 to the local 1-parameter group of local isometrics, we see that each Ni is a closed totally geodesic submanifold. Let x e /Vi ..If we choose a suitable orthonormal basis for T(M), then the linear endomorphism X)), of Tx(M) is given by a matrix of the form

Write T(M) = T + Ti', where T (resp. T:) is the subspace spanned by the first n — 2k elements (resp. the last 2k elements) of the basis of T(M). Then Tic is left fixed pointwise by exp (t A) and hence by exp(t X). It is clear that . 7 is the tangent space T;,(/Vi) and Tic' is the normal space Tx1 (1Vi). This proves (1) and the first statement of (2). According to Proposi-tion 2.2, we have Vy (A x) = R(X, Y) for every vector Yon M. This may be rewritten as

Vy (VX) = — R (X , Y) for Ye T(M).

At every point of Zero(X), the right hand side R(X, Y) vanishes and hence Vy (VX) for every vector tangent to M at a point of Zero(X). This proves a little more than what is claimed in (2).

Since the eigen-values + i al , ..., +ak of (VX)„ defined above remain constant on each Ni because VX is parallel on I , we can decompose the normal bundle Ti(J) into subbundles E1 , , Er as follows:

Ti (lVi)= E1 + - - - + Er (orthogonal decomposition),

where El , , Er correspond to the eigen-values — b?, , — br2 of (VX)x restricted to T1(/Vi). Let J be the endomorphism of T i(Ni) defined by

J I E.=-1

(VX). Then J 2 = — I, and J defines a complex vector bundle bi

structure in Ti (lVi). Since T(M) I N = TOO+ T i (lVi), if T(M) is orientable, T(N1) is also orientable. q. e. d.

Since the fixed point set F of isometrics or the zero set of infinitesimal isometrics consists of totally geodesic submanifolds, it is perhaps appro-priate to mention the following result on totally geodesic submanifolds at this point.


Theorem 5.4. Let N be a totally geodesic submanifold of a Riemannian manifold M. If X is an infinitesimal isometry of M, its restriction to N projected upon N defines an infinitesimal isometry of N. In particular, if M is homogeneous and N is a closed totally geodesic submanifold, then N is also homogeneous.

For the proof, see Kobayashi-Nomizu [1, vol. 2; P. 59], where other properties of totally geodesic submanifolds are also given.

This is probably an appropriate place to mention the following topo-logical result.

Theorem 5.5. Let M be a compact Riemannian manifold and X be an infinitesimal isometry of M. Let Zero (X). H M be the decomposition of U - the zero set of X into its connected components. Then

(1) (- ok dim Hk (M; K)=E(E(— 1)k dim lik (Ni ; K)), t k

(2)E dim Hk (M K)> EE dim Hk(lVi; K)) for any coefficient field K. k

Proof We shall prove (1) following Kobayashi [5]. The proof of (2) is harder and is omitted (see Floyd [1] and Conner [1]).

Let A i be the closure of an s-neighborhood of N . We take e so small that every point of A i can be joined to the nearest point of N, by a unique geodesic of length e and that Ai n Ai be empty for i j. Then A i is a fibre bundle over Ni whose fibres are closed solid balls of radius s. Set A =-- U A. Let B be the closure of the open set M —A. Then A n B is the boundary of A.

We remark that if

Uk Vk '47k 1 -4 Vic — "

is an exact sequence of vector spaces, then

( _ 1)k di m uk _E( _ dim vk _ dim wk = 0

We apply this formula to the exact sequences of homology groups (with coefficient field K) induced by

B - 11/1 — (M,B) and AnB — A —*(A,AnB) and obtain

X (B) — X(M)+X(M, B)=0 and

where x denotes the Euler number. By Excision Axiom, (M, B) and (A, A n B) have the same relative homology. Hence, x (M, B)= x (A, A n B). It follows that

5. Fixed Points of Isometrics 63

Since the 1-parameter group generated by X has no fixed points in B nor in A n B, Lefschetz Theorem implies x(B)= x (A n B) = O. Hence, x(M)= X(A). Since Ai is a fibre bundle over Ni with solid ball as fibre, we have

Finally, we obtain

X(M=E X(Ai)=E X(N)- We should remark that in Floyd [1] and Conner [1] (2) is stated as

follows. If T is a total group acting on a manifold M with fixed point set F, then for any coefficient field K we have

Edimilk (M; K)>EdimHk (F; K).

To see that their statement means (2), let T be the closure of the 1-para-meter subgroup of 3(M) generated by X. Then T is a connected compact abelian group and hence a toral group. Clearly, F =Zero(X). q.e.d.

As a generalization of (1) of Theorem 5.5, we mention the following result. Let M be a compact Riemannian manifold and f be an isometry of M. Let F be the fixed point set off If we denote the Lefschetz number off by L(f) and the Euler number of F by x(F), then

L(f)= X (F)-

To prove this statement, we have only to replace Zero (X) by F and the Euler numbers x (M), x (A), ... by the Lefschetz numbers L(f), L( f IA), .... This result has been proved by Huang [1] when f is periodic.

We shall see that (1) of Theorem 5.5 is a very special case of Theo-rem 6.1 in the next section. It becomes also a special case of the classical theorem of Hopf when the zeros of X are isolated since an infinitesimal isometry has index 1 at each of its isolated zeros.

For various homological results on periodic transformations and toral group actions, see Borel [3] and the references therein.

An infinitesimal version of the following theorem is due to Berger [1]. The generalization given here is due to Weinstein [1]; he proved the result for a conformal transformation. The idea of the proof is similar to that of Frankel [2].

Theorem 5.6. Let M be a compact, orientable Riemannian manifold with positive sectional curvature. Let f be an isometry of M.

(1) If n =dimM is even and f is orientation-preserving, then f has a fixed point.

(2) If n =dim M is odd and f is orientation-reversing, then f has a fixed point.


Proof. Let d(p, f (p)) be the distance between pe M and f (p)E M. It is a non-negative function on M. Let po be a point of M where this function d(p, f (p)) achieves a minimum. We must show that the minimum is zero. Assume that d (Po , f (pa is positive. Let c be a minimizing geodesic from Po to f(p 0).

Let N and N' be the normal space to c at po and f (p o), respectively. We shall show that f maps N onto N'. We consider a 1-parameter family of curves cs , —e<s <e, such that c = co and, for each fixed s, the starting point of cs is mapped into the end point of cs by f If we denote by L(s) the arc-length of cs , then L (0)=0. On the other hand, (see for instance Kobayashi-Nornizu [1, vol. 2; p. 80])

a

11, (0) = g (X , (,,0) — g (X , S g (X , VT T) dt ,

where T is the Vector field tangent to c, X is the variation vector field defined by the family cs and a is the arc-length of c. Since c is a geodesic, the integrand on the right hand side vanishes. The vector field X can be prescribed at po . In particular, let X be perpendicular to T at p . Then L (0)=0 implies that X is perpendicular to T at f (p o). Since Xf(po) =f (Xpo), this proves that f maps N onto N'.

We claim that f maps the initial velocity vector T c to the velocity vector Tf (pi)) of c at f (p o ). Since f maps N onto N', it is clear that f (Tpo ) is either Tf(po) or — Tf (po) . If we choose X in the formula above for (0) in such a way that X,, 0 = Tpo , then g (X, T)f(po) =g (X, T)po = 1, which implies our assertion.

If we denote by A the composition of f: 7p0 (A4).— Tf(po) (M) and the parallel displacement from f (p o) to p o along c (in the reversed direction), then we have a linear automorphism of the tangent space Tpo ( 4) leaving N invariant. According as f is orientation-preserving or orientation-reversing, the orthogonal transformation A (and also its restriction to N) has determinant 1 or —1. If n =dim M is even (resp. odd) and f is orienta-tion-preserving (resp. orientation-reversing), then A leaves a unit vector, say X, of N fixed. We extend this vector to a parallel vector field X along c by parallel displacement. By construction, f (X po)= X pin)) . We define a 1-parameter family of curves Cs , — 8 <s < e, as follows. For each fixed t, we set cs (t)=exp(s X00). Then for each fixed t, C 5 (t) describes a geodesic as s varies and has Xc (,) as the tangent vector at s =O. Let a be the arc-length of c so that f (p 0). c (a). Since f (X 1 X - c (0), == - c (a), it follows that f(c5 (0))=cs (a). Let L(s) denote the arc-length of cs . From the way p o was chosen, it is clear that L(s) achieves a minimum at s =O. Hence, L' O. On the other hand, the second variation 12 (0) of the arc-length is given by a

L' (0). [g (VT X, VT X) — g(R(X, T) T, X)] dt . o

5. Fixed Points of Isometries 65

(In general, terms involving the second fundamental forms of the 1-dimen-sional submanifolds c 5 (0) and c5 (a) must be also taken into account. But they do not appear here since both c.,(0) and c5 (a) are geodesics. See, for instance, Bishop-Crittenden [I; p. 219].)

Since X is parallel, we have VT X = O. By the hypothesis on the cur-vature, we have g(R(X , T) T, X)>0. Hence, the formula above implies If (0) < O. This is a contradiction. q.e.d.

From Theorem 5.6 we may derive the following result of Berger [1 ].

Corollary 5.7. If M is an even dimensional, compact Riemannian mani-fold with positive sectional curvature, then every infinitesimal isometry X of M has a zero.

Proof If X has no zeros, then exp X) has no fixed points for small t. If M is orientable, this is a contradiction by Theorem 5.6. If M is not orientable, let SI be the orientable double covering space of M and let be the infinitesimal isometry of gl induced by X. Then apply the same reasoning to 14 and X. q.e.d.

Remark. The original proof of Berger goes as follows. Set f= g(X, X) and let po be a point where f achieves a minimum. If V is a non-zero vector at po , then V 2 f (V, 0; for the second covariant derivative V 2 f of at po is the Hessian off at p o . On the other hand, V 2 f (V, V) is given by (Lemma 2 for Theorem 4.1)

f (V, V) = g (Vy X, Vy X) — g(R(X , V) V, X).

Assume that X has no zeros, i. e.,f is positive everywhere. We shall find a vector V at po such that V 2 f (V, V) is negative. Consider the linear endo-morphism of the tangent space Tpc,(M) given by VX (= — Ai). We claim that VX annihilates X at po , i.e., (Vx X)p. =O. In fact, for , every vector U at Po' we have 0 = U f=g(Vu X, X)= —g(Vx X, U), the last equality being the consequence of the skew-symmetricity of VX. Hence, (Vx X) p0 = O. Being a skew-symmetric linear endomorphism of Tp.(M), VX is of even rank. Since it annihilates X,, 0 , it has to annihilate another nonzero vector, say V, at po perpendicular to X. Then Vv X =O. Since the sectional curvature g(R(X , V) V, X) is positive, it follows that V 2 f (V, V) is negative. This is a contradiction.

Corollary 5.8. Let M be a compact Riemannian manifold with positive sectional curvature.

(1) If dimM is even and M is orientable, then M is simply connected.

(2) If dim M is odd, then M is orientable.


Proof (1) Let it be the universal covering space of M. Every deck-transformation of Al is orientation-preserving and hence must have a fixed point by Theorem 5.6. This is a contradiction unless M itself is simply connected.

(2) If M is not orientable, let AI be the orientable double covering space of M. Then the non-trivial deck-transformation of it/ is orientation-reversing and must have a fixed point by Theorem 5.6. This is a contradic-tion. q. e. d.

The corollary above is originally due to Synge [1]. From this corollary and Theorem 5.6, we obtain the following result.

Corollary 5.9. Let M be a compact Riemannian manifold with positive sectional curvature. If M is not orientable, then every isometry of M has a fixed point.

Proof By Corollary 5.8, dim M is even, and the orientable double covering space if of M is simply connected. We lift an isometry f of M to an isometry f of By composing it with the non-trivial deck-trans-formation if necessary, we may assume that f is orientation-preserving. By Theorem 5.6,f has a fixed point. Hence, f has a fixed point. q.e.d.

For other applications of Theorem 5.6, see Weinstein [1]. In the case of non-positive curvature, the following theorem of E.

Cartan is basic (see Kobayashi-Nomizu [1, vol. 2; p. 111] for a proof):

Theorem 5.10. Every compact group IB of isometries of a complete, simply connected Riemannian manifold M with non-positive sectional cur-vature has a fixed point.

We should point out that the fixed point set F in Theorem 5.10 is connected. Suppose p and q are two points of F and consider the (unique) geodesic from p to q. Every element of 0 leaves this geodesic pointwise fixed since it leaves p and q fixed. This shows that p and q can be joined by a geodesic which lies in F.

In connection with this, we note that if is a connected Lie group of isometries acting on a complete Riemannian manifold with non-positive positive sectional curvature (not necessarily simply connected), then its fixed point set F is connected (possibly empty, of course). This follows from Corollary 5.2 and from the fact that M is free of conjugate points.

Let M be a complete Riemannian manifold with non-positive sec-tional curvature and ni (M) be its fundamental group. Then each element of ni (M) acts on the universal covering manifold a without fixed point. Hence, the study of the fixed points of an isometry of fi has a bearing on the study of ni (M). In this connection, see Preismann [1 ], Bishop-O'Neill [1], Wolf [4], Yau [1], Gromoll-Wolf [1 ].

6. Infinitesimal Isometrics and Characteristic Numbers 67

6. Infinitesimal Isometries and Characteristic Numbers

Let M be an oriented Riemannian manifold of dimension n =2m. Let P be the bundle of oriented orthonormal frames over M; it is a principal bundle over M with group SO (n). Denote by Q the curvature form on P. Let f be a symmetric form of degree p on the Lie algebra o(n) which is ad(S0(n))-invariant. For the sake of simplicity, denote by f(0) the 2p- form f(Q, f2) on P. Then there exists a unique closed 2 p-form f (Q) on M such that n* (f (0)) =f (Q), where it: is the projection (see, for example, Kobayashi-Nomizu [1; Chapter XII]). The cohomology class defined by f (0) is called the characteristic class defined byf If p=m and M is compact, then the integral of f (Q) over M is called the charac-teristic number defined by f.

Let X be an infinitesimal isometry of M. Let Zero (X)=H - be the zero set of X, where the 1Vt's are the connected components of Zero (X). As we are interested in one iv, for the moment, denote Ni by N. Let 2r be the codimension of N so that dim N =2m-2r. Let 4, be the bundle of adapted frames over N; it is a principal bundle over N with group SO (2 m —2 r) x SO (2 r). (By an adapted frame, we mean an oriented orthonormal frame whose first (2m-2r) basis elements are tangent to N and whose last 2r elements are normal to N.) The curvature form Q restricted to the subbundle Pig of P is of the form

( f40 b_. 2m —2r; 2m

We recall that a tensorial p-form of type ad (SO (n)) is an o(n)-valued p-form cp on P such that

Rnp) = (ad a-l )cp for a e SO (n),

where R. denotes the right translation of P by SO (n) and

cp (Z 1 , , Z p)= 0 whenever Z 1 is a vertical vector.

For example, the curvature form Q is a tensorial 2-form of type ad (SO (n)). Since the covariant derivative VX of X is a skew-symmetric linear endo-morphism of the tangent bundle T(M), it may be also considered as a tensorial 0-form of type ad(S0(n)) on P; this 0-form on P will be still denoted by VX. If we restrict the 0-form VX to P,„ then it is of the form

(

0 0 \ 0 AP

where A is a matrix of order 2r, see § 5.


Let f be an ad(S0(n))-invariant symmetric form of degree p on the Lie algebra o(n). Consider the form

f (t Q +V X)

on PN. If we denote by D the exterior covariant differentiation (Koba-yashi-Nomizu [1, vol. 1; P. 77]), then not only DO =0 but also D (V X) = 0 on pN since VX is parallel on N as we have shown in Theorem 5.3. It follows that f(tt2+VX) is closed. Denote by f (tC2+ VX) the form on N defined by 7t* (f (t Q +V X)) =f (t Q + VX). We may write

f (t + v x). E f (s-2 v x . . . v tk k k

p - k

The coefficient of tk is a closed 2k-form on N; it is a polynomial in the curvature form with constant coefficients since VX is parallel on N.

Consider now the polynomial det(S), Se o(2p), on the Lie algebra o(2p). We know that the determinant of a skew-symmetric matrix S is a square of a polynomial. More precisely, there is a unique polynomial Vdet(S) on o (2p) such that Wdet (S)) 2 = det(S) and

0 si —si 0

Vdet(S) = s i s2 sp for S.

In fact, (cf. Kobayashi-Nomizu [1, vol. 2; p. 304])

1 Vdet (S) = E i2p 421

422 ; - I for

2" p

where c i2p is the sign of the permutation (1, , 2 p)—* (ii ,, set

1 X

P(S)= (2 iry, (Vdet (S)) for Se o (2 p) .

We know that the curvature form Q restricted to PN splits into the tangential and normal parts. We denote by Qv the normal part (Qii), i,j= 2 m-2 r +1, . . . , 2 m. We denote the normal part of VX by A as before. Since the form xr (tS2,+ A) on PN is defined by an ad(S0(2 r))- invariant polynomial xr on o (2 r), there is a unique closed form kr (t Qv +A) on N such that n*(27,(t + = Xr(t 2 v ± A). Since VX is parallel on N, )7,(t Qv + A) expands into a polynomial in t whose coefficients are all polynomials in Q, i,j>2m —2r, with constant coefficients. Since


det (A) +0, the constant term of .L.(t0,+ A) is nonzero. Hence,

1/( -6.(t 0, + A))

can be expanded into a power series in t whose coefficients are forms on N which can be written as polynomials in 01, i,j> 2 m-2 r.

Let f be an ad(S0(2m))-invariant symmetric form of degree m on o(2 m). Then the residue Rest (N) is defined by

_ = f3_6(:(Qtr2+,, ±VXA))

Resf (N) •

We shall now prove the following formula of Bott [1, 2]. See also Baum-Cheeger [1].

Theorem 6.1. Let M be a compact, oriented Riemannian manifold of dimension n =2m. Let X be an infinitesimal isometry of M with zero set Zero(X). H where the 1Vi 's are the connected components of Zero(X). — Let f be an (ad SO(n))-invariant symmetric form of degree m on o(n). Then the characteristic number (Q) of M defined by f is given by

U()=> Resf (1Vi). pgi

Proof The first formula we are going to prove is the following:

( 1 ) tdf(tfl-EVX)=L x (f(ti2+VX)), on M,

where tx is the interior product by X. We shall do all our calculations on the principal bundle P so that X and VX are also lifted to P in a natural manner. We denote by•D the exterior covariant derivation. Then

td(f(tS2+VX))=tD(f(ti2+VX))=tD(f(t0+VX,...,t0+VX))

=mtf(DVX,t0+VX,...,t0+VX)

=mtf(t x f2,t0+VX,...,t0+VX)

=mtf(t x (tS2+VX),t0+VX,...,ti2+VX)

=t x f(tfl+VX).

In the proof above of (1), we made use of the formula DV X =ix 0; this formula is equivalent to the formula in (1) of Proposition 2.2 but can be derived easily and directly from Lx w=0 and t x co =DX, where co is the connection form.

We define a 1-form on M-Zero (X) by

( Y) = g (X, Y)/g (X, X) for Ye T (M-Zero (X)).


Then

(2) 1/J(X)=1, Lt/,=O and i1 ch/J=0.

We set n=1 (tfl+VX) 1—t

where 1/(1— tchk) means 1 +tdiii + t 2 (d 111) 2 + We prove the following formula:

(3) AtS2+VX)+tdn—i x ti=0.

This is a consequence of (1) and the following two formulae:

tdri=tdf(tO+VX) tfr +hi-2+DX) tchk 1 —t 1—tdtk

1 ixn=ixi(to+vx) 1---t' d

+1(112 +VX) 1—t di '

which follows from (2). We are now interested in the coefficient of r in the formula (3). We

may write ±

where tik is a (2k + 1)-form. (Since dim M =2m, the coefficient of ti‘ vanishes if k is greater than m). Then

txtl=txt/m-Ir -1 ±••••

On the other hand, the coefficient of r in f(tf2+VX) is given by f(C2). Hence, we have

(4) f(Q)+dtim - 1 .0 on M-Zero(X).

Let N denote the 6-neighborhood of N. Using (4) and Stokes formula, we obtain

If(Q)=1im S f(Q)= —lim E-■ M—UNic E-, M—UNic

.Etim S nm- 1• E_pome

To complete the proof of the theorem, we have to show that

Resf (lVd=lim e.o

and


Since we are now interested in each individual Ni , we denote Ni by N. The boundary NV, of the e-neighborhood of N is a sphere bundle over N with fibre S2 '- ', where 2r is the codimension of N. Let

CE : aivs—,N

be the projection. If we denote by ooNj and 0 (N) the algebras of differential forms on aNE and N, respectively, then we have a natural algebra homomorphism

a:: 0 (N) - 0(3Nc).

On the other hand, the integration over the fibre in aisle - q■1 will be denoted by al . Hence

ce* : 4, 03 NE) -* 0(N)

is a linear mapping which decreases degrees by 2r - 1. Moreover, as. is characterized by the following formula:

S u • a: v = I a; u • v for u e (alve), v e 0 (1V) . ON , N

If cp is a form defined on M-Zero (X), then we denote by afi, ((p) the integral over the fibre of the restriction of cp to a1\. We set

E.o provided the limit exists.

To calculate the integral lim s n„, -1' E-.0 ai s T.

we first integrate ri„,_ 1 over the fibre in aNs —)N and then integrate the result over the base N. Hence, we are interested in o-* (ri 1 ). But this is the coefficient of tm-' in a. ?I. We are now interested in calculating

tfr \ =0-* (f (t Q +V X) 1 — t dill I .

Since f (t C2 +DX) is smoothly defined on the entire space M including N, it follows that

til (5) a* n=f(to +vx)IN . a* (1 — t dill I

\ '

The problem now is to evaluate a* (Oil - t d 114 Since

Ill 1-tdtk

=tp+tikAdt/J-Ft 2 tfrA(dt/4 2 +•••,

the problem is further reduced to that of calculating a* (4/ A (d Or 1.


Let be the 1-form corresponding to the infinitesimal isometry X under the duality defined by the metric tensor. Then

= 2 IIII '

and

di I i . g0 2 g —dC112) A

g 11 4 so that

(6) tii A (d tlir - 1 = gii 2 4 •

, We fix a point o of N and let x1 , ..., x2 p, y', ..., y2 r m=p+r, be a normal coordinate system around o such that N is locally defined by yi ....=y 2 r=0. Such a normal coordinate system exists since N is totally geodesic. We consider the Taylor expansions of and cg at o. Then

(7) =E A u y i dyi+•••,

(8) g=EA u dy i Adyi-Ekiab A ik yk yi dxa Adx b +--,

where the dots indicate terms with total degree in y and dy greater than 2 and (At,) is a skew-symmetric non-degenerate matrix. These two formulae may be proved as follows. If we denote yi. , ... , y2 r by x2 P+1 , ..., x2 p +2 r

and write =E A dx A , then

(A)=O

4; B± B; A" (2) of Proposition 2.2,

&4;B;C+ E RAB D= 0 (1) of Proposition 2.2,

(41)0 =0 unless A, B._., 2 p+ 1 (Theorem 5.3),

(&4;B)A,B=2 p+1,...,2 p+2r is non-degenerate at o (Theorem 5.3), (2 p

E RO a b dxa A dxb) =0 unless C, D -2p a, b=1 o

Or C, D 2p +1 (Nis totally geodesic).

From these formulae, both (7) and (8) can be easily obtained. It is clear that in calculating a* (lk A (4)4-1 ) we can replace and cg by their Taylor expansions and ignore the terms of degree sufficiently high in y and dy. It is indeed not difficult to see that the terms indicated by dots in (7) and (8) can be ignored. (For the detail on this point, see Bott [2; pp, 321-323 ].)

6. Infinitesimal Isometries and Characteristic Numbers 73

We set =E Aii y i d yj,

13=EAu dy 1 dyi,

y=ER; ab A ik yk yi dxa Adx b

so that and d c are approximated by a and # — y, respectively. From (6), we obtain

(

a A (fi 0 04 2 11 I

Since the dimension of the fibre in aivE -3 N is 2r — 1, the integration over the fibre annihilates any term whose degree in d y is not exactly 2r —1. This reduces (9) to

o.* (1// A (d o)"-1)=0 for q <r,

(10) 0-* (0 A(4)41-1)='( q -ri * k 11Œ11 24 for q r.

Hence,

a* ( 1 —1kt a* (E tg- 111 A ((I "- 1)

(11) =a (qcr

(q

_1 œ A fir-11 1:11‘2(4—Y)_r

=a ((DOE:11213+7; 1 0r tr )

We set z,=EA ijyi,

Cu =ERiab zei d x° A dxb,

where (A4) is the inverse matrix of (A). (In the sequel, by fixing d xa we consider (0) as a constant matrix rather than a matrix valued 2-form at o. In other words, di means actually 0-1 (U, V) where U and V are arbitrarily chosen tangent vectors of N at o.) We claim that (0) is sym-metric, i.e., Ci= d i . In fact, since the 1-parameter group exp((tVX)0) of linear transformations of TAM) preserves the curvature tensor at o, we have

Rik a b Aki=EAik Rkjab for a,b2m —2r and i,j,k>2m —2r.

Our assertion follows from the fact that Aik = — A" and Rkiab= —Ri kab• Now we can write 11Œ112 and y as follows:

ce= 11z11 2 (=Ezizi),

y = C(z, z) ( = E cif z, zj).

(9)


Choosing y', ... , y2 " in such a way that il ii =0 except for i = 2 s — 1, j= 2 s or i = 2s, j = 2 s— 1 where s = 1 , . . . , r, we can easily verify the formula:

2r- ' Œ A fir -1 = — (r — 1)! v (z) ,

(-1/1) where

A =(Aii), v(z)=D—iydz i A •-• A dz i _ i A z i dzi+i A - • A dz 2r . i

Then ŒA fir-1

(12) (r (P(Z; t)

Oa II 2 ± Y 02 1)! 2r -1

= -V det (A) ' where

y(z) (p(z;t)=

I 1z112

+ C(Z, Z) t .

For each fixed t, p(z; t) is a (2 r — 1)-form which is defined and closed on R2 r —0. It follows that, for any closed hypersurface H in R2 r —0 homo-topic to the unit sphere Ilz II =1,

(13) f (p(z; t)= 1 cp(z; t) . H Ilz II = 1

We set h(t)= f cp (z; t).

Ilz II = 1

If t is fixed to be a sufficiently small constant, then the quadratic form 11z11 2 ± C(Z, Z)t is positive definite. Hence, there exists a linear automor-phism A, of R 2 ' such that tA, A 1 = I + C t, where I denotes the identity matrix of order 2 r. We may impose even the condition that A, be orienta-tion-preserving, j. e., det (20> 0. If we set

w=i1,z, then

v(w)=det (A t) • v (z) ,

11z11 2 + C(z, z)t =11w11 2 so that

- v(z) v (w)

niat zsu - i 014 2 + qz, z) tr = li wif=. 1 det (4) • 11w11 2r .

But the left hand side is equal to h(t) by (13). Hence,

1 (14) h(t)= 1 v(w) .

det (At) Il w II = 1

6. Infinitesimal Isometries and Characteristic Numbers 75

From the definition of A„ we obtain

det(A1 ) 2 = det (/ + C t) .

We denote (Au) by A and (E Rijab d x° A d xb) by Q v so that (A ii)= ii - 1 and (Cii)= Qv . A'. Then

det (I + C t)= det (A - 1 ) det (A + Q , t)

=(Xr(A) - 1 Xr(A + Qv t)) 2 .

Since det(A) is positive, we can conclude that

(15) ' det (Ad= Xr(A) - 1 Xr (A +Q t) .

By Stokes formula, we obtain

S y (w) = f d (I) (w)) II w II - 1 !hull .1-

(16) nr = $ (-.2r)dw1 Adw2 A — Adw2 ,.=(-20 37

!.

IlwIll

From (13) and the definition of h(t), we have

(17) a* (cp(z; 0)=1im S (p (z; t)= 1 cp(z; t)= h (t) . E. o pH „ Hz II = 1

From (11), (12) and (17), we obtain

(18)(r — I)! 21* - 1

h (t) . a* (1 !it dik ) = tr- 1 c-{10

From (14), (15), (16) and (18), we obtain

(19) 'Il '1 tr-1 a* (1—tchk)

=MA+t(2,) '

From (5) and (19), we obtain

(20)f(tr2+VX)IN

c * 11 = -if. (1. Q v + A)

Since 1m -1 is the coefficient of tm - ' in n, it follows from (20) that a * (tr -1 ) is the coefficient of en - r in

f (t 0 +V X)IN

Hence, f( t C2 +V X)

lEi101 .1. n m - 1 tm- r = N Sa*(tini-1)tm-r= 1 Xr(tOv+A) . ON6


But the right hand side is equal to Resf (N)rn - r by definition. Hence,

lim Jn.- 1 =Res1 (N). £-° Ci ON,

This completes the proof. q.e.d.

If N is an isolated point {p}, then

AVM

Resf (p)= Xrn (A)

.

This means that if Zero(X) consists of isolated points only, then the characteristic numbers of M can be expressed in terms of VX (=A) at these isolated zero points. Even when an invariant polynomial f is so chosen that f(S2) represents an integral class, the individual Resf (p) need not be an integer. It is difficult to compute Resf (p) unless M is a symmetric space. If Zero(X) is empty, then the right hand side of Theorem 6.1 vanishes. Hence,

Corollary 6.2. If a compact, orientable Riemannian manifold admits an infinitesimal isometry with empty zero set, then its Pontrjagin numbers vanish.

The proof of Theorem 6.1 given here yields the following result. Let G be a Lie subgroup of SO(2m) and P be a G-structure on a 2m-dimen-sional manifold M. Assume that there is a torsionfree connection in P. Then Theorem 6.1 is valid for an infinitesimal automorphism X of the G-structure P and for an ad (G)-invariant polynomial f on g of degree m. Note that, since Gc SO(2m), an infinitesimal automorphism X of P is an infinitesimal isometry but that an ad(G)-invariant polynomial on g may not be induced from an ad(S0(2m))-invariant polynomial on so(2 m). In particular, Theorem 6.1 is valid when G = U (m) and P is a Kdhler structure on M.

For a completely different proof of Theorem 6.1, see Atiyah-Singer [1].

III. Automorphisms of Complex Manifolds

1. The Group of Automorphisms of a Complex Manifold

Let M be a complex manifold and j (M) the group of holomorphic trans-formations of M. In general, 5(M) can be infinite dimensional. For instance, 5 (V) is not a Lie group if n 2. To see this, consider transfor-mations of C2 of the form

Z =Z

w' = w +f (z) (z, w)e C 2 ,

where f (z) is an entire function in z, e. g., a polynomial of any degree in z. The fact that 5(0) contains these transformations shows that 5(0) cannot be finite dimensional. Similarly, for 5(e) with n 2. On the other hand, 5 (c) is the group of orientation preserving conformal transformations and, as we shall see later, it is a Lie group. The purpose of this section is to give conditions on M which imply that 5 (m) is a Lie group.

Theorem 1.1. Let M be a compact complex manifold. Then the group 5 (M) of holomorphic transformations of M is a complex Lie transformation group and its Lie algebra consists of holomorphic vector fields on M.

Proof From Corollary 4.2 of Chapter I, we know that 5 (M) is a Lie transformation group. Its Lie algebra can be identified with the Lie algebra of holomorphic vector fields; if Z=X+iX is a holomorphic' vector field with X and Y real, then X is an infinitesimal automorphism of the complex structure of M, and vice versa. So the Lie algebra of 5 (M) is a complex Lie algebra; if Z is a holomorphic vector field, so is .i Z. In other words, if X is an infinitesimal automorphism of the complex structure, so is J X. Hence, 5 (NI) is a complex Lie transformation group. q. e. d.

Theorem 1.1 is due to Bochner-Montgomery [2, 3]; they have actu-ally shown that the topology of 5(M) is the compact-open topology.

78 III. Automorphisms of Complex Manifolds

We consider now the following theorem of H. Cartan [1, 2] :

Theorem 1.2. Let M be a bounded domain in CI. Then the group 5(M) of holomorphic transformations of M is a Lie transformation group and the isotropy subgroup 5„(M) of 6(M) at any point xeM is compact. If X is in the Lie algebra of 6(M), then .1X is not in the Lie algebra of 6(M).

We shall elaborate a little on the last statement. If X is an infinitesimal automorphism, then J X is also an infinitesimal automorphism. In other words, if Z is a holomorphic vector field, then iZ is obviously also holo-morphic. In order that X belongs to the Lie algebra of 5(M), X must be complete, i.e., must generate a global 1-parameter group of global holo-morphic transformations. The last statement in Theorem 1.2 means that if X is complete, then J X cannot be complete. This should be viewed in contrast to Theorem 1.1, where M is compact so that every vector field is complete.

We shall now give two results each of which generalizes Theorem 1.2. We recall first the definition of Bergman metric. Let M be an n-dimen-sional complex manifold and H the complex Hilbert space of holomorphic n-formsf which are square integrable in the sense that

in2 f f < co.

The inner product of H is given by

g)= f r2 fA g

We assume that H is very ample in the following sense: (1) At each point x of M, there exists an feH such that f(x)#0.

(2) If z i , z" is a local coordinate system in a neighborhood of a point x eM, then, for each j, there exists an element

h=h*dz i •••Ade

of H such that h(x)= O and (ah*/azi)x #0. Let ho , h1 , h2 , be a complete orthonormal basis for the Hilbert

space H and define the Bergman kernel form K by 00

K=K*dz i n•••AdznAdI l A• , •Adr.Lh k Ah k . k=0

(If M is a domain in C' with natural coordinate system z1,...,e, the function K* is the classical Bergman kernel function of M.)We define the Bergman metric ds 2 of M by

d S 2 = 2 E d zg • de , where gOEff =a2 log Kiiazaa-±0. co3.1

1. The Group of Automorphisms of a Complex Manifold 79

It is not hard to see that K is defined independent of the choice of ho , h1 , h2 , ... and the metric ds 2 is independent of the choice of zi, ..., e. We remark that Condition (1) guarantees that K*0 everywhere, j. e., K* >0 everywhere so that ds 2 is defined and Condition (2) implies that ds 2 is positive definite. (Without (2), ds 2 is, in general, positive semi-definite.) A more geometric interpretation can be given to (1) and (2) as follows. For each point x of M, let H(x) denote the subspace of H consisting of those holomorphic forms f vanishing at x. Condition (1) says that H(x) is a hyperplane in H. The set of all hyperplanes in H forms a complex projective space (possibly of infinite dimension). Since this projective space is isomorphic, in a natural manner, to the projective space of com-plex lines in the dual space H* of H, we denote it by P(H*). Then we have a mapping A 1 —) P(H*) which sends x into H(x). From the definition of the Fubini-Study metric, it follows that ds 2 is induced from the Fubini-Study metric of P(H*) by the mapping M—)1)(11*). Condition (2) says that the mapping M —, P(H*) is an immersion. For more details, see Kobayashi [6].

Theorem 1.3. Let M be a complex manifold of dimension n such that the space of square-integrable holomorphic n-forms is very ample ( so that the Bergman metric is defined). Then the group .5 (M) of holomorphic trans-formations of M is a Lie transformation group and the isotropy subgroup bx (M) of 5 (M) at any point xeM is compact. If X is a nonzero element of the Lie algebra of .5 (M), then J X is not in the Lie algebra of $ (M) provided one of the following three conditions is satisfied:

(a) There is no parallel vector field ( with respect to the Bergman metric) on M.

(b) There is no holomorphic mapping of C into M except the constant mappings.

(c) There is a point of M where the Ricci tensor is non-degenerate.

Proof Clearly, $(M) is a closed subgroup of the group 3(M) of isometries of M with respect to the Bergman metric. It follows from Theorem 1.2 of Chapter II that $3(M) is a Lie transformation group and bx (M) is compact.

Lemma. If X and J X are infinitesimal automorphisms of a Keihler manifold M, then X is parallel.

Proof of Lemma. We recall the definition of A x (cf. § 2 of Chapter II):

A x =--L x —Vx .

JoA x =A x 0J=Ajx , Then


where the first equality follows from the definition of A x and the second equality is a consequence of the formula A x Y = — Vy X in Proposition 2.1 of Chapter II. By Proposition 2.2 of Chapter II, we have the equation

g(A jx JY,Z)+ g (J Y, Ajx Z)= 0 for all vector fields Y, Z,

which can be easily transformed into

—g(A x Y,Z)+g(Y,A x Z)=0.

On the other hand, from Proposition 2.1 of Chapter II we have

g(A x Y,Z)+g(Y,A x Z)=0.

Hence, g(A x Y,Z)=0 for all Y, Z, and, consequently, A x =0. Since A x Y = —Vy X, this shows that X is parallel, thus completing the proof of Lemma.

To complete the proof of Theorem 1.3, assume that the Lie algebra of 5(M) contains both X and JX. From Lemma it follows that if (a) is satisfied, then X=0. Supposé (b) is satisfied. Since X and JX commute and both generate global 1-parameter group of holomorphic transfor-mations of M, they generate a 1-dimensional complex Lie group acting holomorphically on M. Since every 1-dimensional complex Lie group has C as the universal covering group, we obtain a holomorphic action of C on M. By (b), this action is trivial and, hence, X=0. Suppose (c) is satisfied. If X*0, then X is parallel and M has a flat factor in its de Rham decomposition (locally). This would imply that the Ricci tensor is degenerate. q.e.d.

Remark. It is clear that if M is a bounded domain in Cn, then (b) is satisfied. It is not known if (a), (b) or (c) can, be removed in the theorem above. We mention two important cases where (c) is satisfied:

(1) M is homogeneous, i.e., 5(M) is transitive on M. (2) There is a discrete subgroup I' of 5(M) acting freely on M such

that the quotient manifold MIT is compact. (In particular, the case where M is compact is contained in this case.)

To see that (c) is satisfied in the two cases above, let

V=V*dz i n•••AdznAdVA.••/s (I?

be the volume element on M defined by the Bergman metric ds 2 . We recall that the components of the Ricci tensor are given by

ROEff = -- a2 log rivazaav.

We compare this with the definition of the components &co of the Berg- man metric. If M is homogeneous, an invariant volume element is unique

1. The Group of Automorphisms of a Complex Manifold 81

up to a constant factor so that 17*= c K*, where c is a nonzero constant. Hence, Ro= .—go . In case (2), K and I/ can be considered as 2n-forms on M/T since they are invariant by E Then the 2-forms

1 1 2ni Eg

ŒodeAdg and

2 i ER,co dzŒndV

n

on M/T define the same cohomology class, the first Chem class cl (M/T), of M/T. If we use go, we see immediately that ci (M/r)"*O. On the other hand, if det(Ro)=0 everywhere, then cl (M/fr =0, which is a contradiction.

To state another generalization of the theorem of H. Cartan, we define a certain intrinsic pseudo-distance on a complex manifold M. Let D be an open unit disk in C with Poincaré distance (i.e., non-Euclidean distance) p. Given two point p and q of M, we choose a sequence of points P r-- Po , Pi, • • • , Pk-1, Pk =q in M, points al. , ..., ak , b 1 , ...,bk in the disk D and holomorphic mappings fl , ...,fk of D into M such that fi (ai) = f _ 1 (b i _ i )= pi _ i for i = 1, 2, ... , k and fk (bk )= pk . We set

k dm (p, q)= inf ( E p (a i , b i)) ,

i.i.

where the infimum is taken with respect to all possible choices for p i , ai , b i , fi above. Then dm is a pseudo-distance; it is symmetric and satisfies the triangular axiom. For details on this pseudo-distance, we refer the reader to Kobayashi [9, 10]. If dm is a distance, then M is called a hyper-bolic manifold. A hyperbolic manifold M is said to be complete if dm is a complete distance.

Theorem 1.4. Let M be a hyperbolic manifold. Then the group 6 (M) of holomorphic transformations of M is a Lie transformation group and the isotropy subgroup 6„ (M) of 5 (M) at any point x eM is compact. If X is a nonzero element of the Lie algebra of b (M), then J X is not in the Lie algebra of

Proof We make use of the following basic property of dm .

Lemma. If M and N are complex manifolds and f: M--) N is a holo-morphic mapping, then

dN (f (p), f (q))<d m (p, q) for p, q e M .

This lemma follows immediately from the definition of dm and dN . In particular, if M =N and f: M— )M is a biholomorphic mapping,

then f is an isometry with respect to dm . Hence, 6 (M) is a closed subgroup of the group 3(M) of isometries, which is known to be a Lie group by Theorem 3.3 of Chapter I and Theorem 1.1 of Chapter II. It follows that


$ (m) is also a Lie group. By Theorem 1.1 of Chapter II, 5„ (m) is compact. Assume X is a nonzero element of the Lie algebra of .5 (M) such that J X is also in the Lie algebra of (M). Then X and J X generate a 1-dimensional complex Lie group. Taking its universal covering group, we may assume that the group C acts on M. For each point p of M, we consider the orbit of the group C through p. In this way, we obtain a holomorphic mapping of C into M. But the pseudo-distance de on C is trivial, i.e., 4=0. It follows from Lemma that every holomorphic mapping of C into M is a constant mapping. Hence, the orbit of C through p reduces to the single point p. This means that C acts trivially on M, i.e., X is the zero vector field on M. q.e.d.

From the differential geometric standpoint, the most interesting example of hyperbolic manifold is given by a hermitian manifold with holomorphic sectional curvature bounded above by a negative constant (Kobayashi [10]). For results essentially equivalent to Theorem 1.4, see also Kaup [4], Wu [1]. For more details on holomorphic transformations of bounded domains, see Kaup [1].

For a generalization of Theorem 1.1 to compact complex spaces (with singularities), see Gunning [1], Kerner [1]. For generalizations of Theorem 1.2 and 1.4 to complex spaces, see Kaup [1, 4]. H. Fujimoto [1] unifies all these generalizations. For automorphisms of special domains (homogeneous bounded domains, Siegel domains), see Pyatetzki- Shapiro

[1], Kaup-Matsushima-Ochiai [1], Kaneyuki [1], Tanaka [8]. In connection with Theorem 1.2, for the case where the Bergman

kernel form is positive but the Bergman metric is only semi-positive, see Lichnerowicz [5]. For automorphisms of a complex manifold with volume element, see Koszul [1].

2. Compact Complex Manifolds with Finite Automorphism Groups

It has been known for a long time that the automorphism group of a compact Riemann surface of genus greater than 1 is a imite group, Klein (see Poincaré [1]), Hurwitz [1]. In this section, we shall generalize this classical result to higher dimensional compact complex manifolds. The first generalization is the following (Kobayashi [7]).

Theorem 2.1. Let M be a compact complex manifold with negative first Chern class. Then the group 5 (M) of holomorphic transformations of M is finite.

Before we proceed with the proof of the theorem, we shall elaborate on the assumption of "negative first Chern class". We say that the first Chem class c 1 (M) of M is negative if it can be represented by a

2. Compact Complex Manifolds with Finite Automorphism Groups 83

closed (1, 1)-form E yap de d?

2 n

such that (y) is everywhere negative definite. If 2 E gap dz dz° is a hermitian metric on M, then c1 (M) can be represented by

2 E Rap de A dYfl, where Rap = — a2 log (det (g), 5))/azŒ

This shows that a hermitian manifold with negative defmite Rica tensor has negative first Chem class. The 2n-form

in2 Gdz i A • • • A de A A • • • A G= det(gail),

is the volume element of the hermitian metric 2 E &if de difl. More generally, if

in2 lid? A • • • A dzn A d 1 A • • • A din

is any volume element of a compact complex manifold M, i. e., if V> 0, then the 2-form

E yŒp de A dYfl, where yŒp = —a2 log viazaav, 2n

represents ci (M). In order to prove Theorem 2.1, we have to reformulate the assumption

of "negative first Chem class" in algebraic terms. Let K denote the canonical line bundle of a compact complex manifold M; by definition, a local holomorphic section of K is a locally defined holomorphic n-form (where n =dim M). The line bundle K is said to be ample if there exists a positive integer p such that the line bundle K P = KO • • • OK is very ample in the following sense. Let H be the space of holomorphic sections of KP; H = (M ; Ks'). At each point x of M, consider the subspace H(x) of H consisting of sections vanishing at x. Then the condition is that H(x) is a hyperplane of H for each x and the mapping x —) H(x) gives an imbedding of M into the complex projective space P(H*) of hyperplanes in H. (Since the hyperplanes in H are in a natural one-to-one correspondence with the complex lines in the dual space H* of H, the notation P(H*) is justified.) We recall that the Bergman metric exists on M if K itself is very ample (see § 1).

We claim that the canonical line bundle K is ample if and only if c, (M) is negative. The implication "K ample ci (M)< 0" is trivial. Let

coN be a basis for H. If we write formally

coo ro0 + • • • + CON C0N= V (dzi A • • • A dz n)" • (di' A • • • A di",

8,4 III. Automorphisms of Complex Manifolds

then the 2-form

• 1 2n i

E yo de /wig , where yccp = — a2 log v/azaa?

represents the characteristic class of ICP, which is equal to —p • c1 (M). A simple local calculation shows that if IQ' is very ample, then (yap-) is negative definite. The implication " c 1 (M)< 0 —) K ample" is a result of Kodaira [2] and will not be proved here.

In the proof of Theorem 2.1, we take "K ample" as our definition of "cl ()4)<0".

Proof of Theorem 2.1. Let p be a positive integer such that ICP is very ample and let coo , ..., cox be a basis for H = H° (M; IQ). Then the mapping

t : x —) (co0 (x), ... , coN (x)) x eM

defines an imbedding of M into PAC). (Although co0 (x), . . . , coi., (x), are not numbers, their ratio makes sense and defines a point of PN (C)). Every holomorphic transformation cp of M induces a linear transforma-tion of H which will be denoted by p Op). We denote by a(p) the pro-jective transformation of P,., (C) induced by p (çp). Then

a (çp) o t=to (p.

In other words, the imbedding t: M —)PN (C) allows us to represent the group b(M) of holomorphic transformations by a group of projective transformations of PN(C). It is clear that both p and a are faithful re-presentations.

Lemma 1. The image a(6(M)) of a consists of exactly those projective transformations of PN (C) which preserve t(M).

Proof of Lemma 1. Let T be a projective transformation of PN (C) which preserves t(M). Let cp be the restriction of T to t(M). Since t is an imbedding, cp can be considered as a holomorphic transformation of M. Now it suffices to show that T = a (çp). Since a (çp) • T-1 is a projective transformation of PN (C) which induces the identity transformation on t(M), we have only to show that if T is a projective transformation of Ph, (C) which induces the identity transformation of t(M), then T is the identity transformation of PN (0 . Let T be such a projective transformation and ' r a linear transformation of H which induces T We shall show that t= c I, where c is a constant and I is the identity transformation of H. From To t= t and from the definition of t, it follows that

Cr co) (z) = c (z) • co (z) for co e II and zeM,

where c(z) is a nonzero complex number which is independent of co. Since both co and t CO are holomorphic sections, c(z) must be also holo-


morphic in z. As M is compact, c (z) must be constant. This completes the proof of Lemma 1.

Lemma 1 implies that a(S3(M)) is a closed subgroup of the projective transformation group of PN (C). It shows also that a(5 (M)) is an algebraic group.

We shall now construct a bounded domain in H which is invariant by the group p (6 (M)). To this end we introduce a real valued function I, on H, which is very much like a norm. Every holomorphic section of the canonical line bundle K is a holomorphic n-form on M. Hence, every element co of H = H° (M; KP) can be symbolically written locally as follows: co =f . (dz i A .. • A de)P,

where f is a holomorphic function defined in the coordinate neighborhood in which z i. , ..., zn are valid. We define

v(co)--= j i"2 . ( ft) 1/P • dz i A • • • A d? A de A • • • A dYn . m

Then v(w) is well defined, independently of the choice of local coordinate system. The following lemma is trivial.

Lemma 2. (1) v(co) : 0, and v(w)=0 if and only if w= O; (2) v(c w) Ici v(co) for ceR;

(3) v is a continuous function on the finite dimensional vector space H;

(4) v((p* co)=v(co) for cp eH(M).

We define now a bounded domain in H.

Lemma 3. The open subset D of H defined by

D={coeH;v(w)<1}

is a star-like bounded domain invariant by 5(M).

Proof of Lemma 3. By (2) of Lemma 2, every point of D can be joined to the origin by a straight line in D, showing that D is star like and, in particular, connected. To see that D is bounded, let coo , .. be any basis for H. Let S 2N + 1 be the unit sphere in H defined by

s2N+1 ={E aicoi; Elaii 2 =1).

Let vo be the minimum value of the function v on 5 2N + 1 ; since v is continuous, yo exists and, by (1) of Lemma 2, must be positive. Let r be a positive number such that r 2/P v o > 1. Then, by (2) of Lemma 2, D is contained in the ball B defined by

B--={Eai coi ;Elai l 2 <r2}.

Finally, the invariance of D by p(H(M)) follows from (4) of Lemma 2. This completes the proof of Lemma 3.


We shall now complete the proof of Theorem 2.1. Let 6 be the group of linear transformations of H leaving the bounded domain D invariant, where D is defined in Lemma 3. From Theorem 1.2 it follows that 6 is a compact Lie group. On the other hand, since $(M) is a complex Lie group and the action 5(M) x M —) M is holomorphic (by Theo-rem 1, 1), the representation p: (m)-) GL (N + 1; C) is holomorphic. Since p(5(M)) is contained in the compact subset 6 of GL (N + 1; C) ceN + 1)2, p maps the identity component of 5(M) into the identity element of GL (N + 1; C). Since p is faithful, 5(M) is discrete. If we denote the natural homomorphism GL (N +1; C) PGL(N ; C) by 7r, then a=nop and a(5(M)) is a closed subgroup of the compact group n(6) by Lemmas 1 and 3. Hence, cr(5(M)) is compact. Since 5(M) is discrete and a is faithful, 5(M) is finite. q. e.d.

Remark. In the course of the proof, we have established that if M is a compact complex manifold, the identity component of 5(M) leaves every holomorphic section of K", (p 0), fixed (whether K is ample or not).

The second generalization is the following (Kobayashi [9,10], Wu [1]).

Theorem 2.2. Let M be a compact hyperbolic manifold. Then the group $(M) of holomorphic transformations of M is finite.

Proof Let dm be the intrinsic distance defmed in § 1. As we saw in the proof of Theorem 1:4, the group 5(M) is a closed subgroup of the group' 3 (M) of isometrics of M with respect to the distance dm . Since M is compact, 3(M) is compact (see Theorem 1.1 of Chapter II) and hence 5(m) is also compact. It suffices therefore to prove that the identity component of 5 (M) reduces to the identity element. Assume that dim 5(M) is positive. Since 5(M) is a complex Lie group by Theorem 1.1, it is generated by complex 1-parameter subgroups. It suffices therefore to show that the group C cannot act holomorphically on M except in a trivial manner. Let f: CxM be a holomorphic action of C on M. For each fixed p -É-M, the mapping a EC f (a, p)EM is holomorphic and hence is distance-decreasing with respect to de and dm (see Lemma in the proof of Theorem 1.4). Since de is identically equal to zero, this means that d m ( f (0, p), f (a, p)) = 0 for all elements a EC. Since f(0, p)= p and dm is a distance, we may conclude that f (a, p)= p for all a EC. q.e.d.

Examples. If M is a compact Kahler manifold with negative definite Ricci tensor, then c l (M) is negative and, by Theorem 2.1, the group 5(M) of holomorphic transformations is finite. On the other hand, if M is a compact hermitian manifold with negative holomorphic sectional curvature, then M is hyperbolic (Kobayashi [9, 10]) and, by Theorem 2.2, 5(M) is finite. If M is a complete intersection submanifold of r non-singular hypersurfaces of degrees al , , a, in P,i+ ,(C) such that n+ r+ 1


>al + • • • + a„ then e l (M) is negative (see Hirzebruch [1; p. 159]) and, by Theorem 2.1, the group 5(M) is finite. In particular, if M is a non-singular hypersurface of degree greater than n +2 in Pn+1 (C), then 5(M) is finite. It is of some interest to note that the hypersurface M in Pn+i (e)

n+1

defined by E (z )" =o in terms of a homogeneous coordinate system i o

z°, .. . , f + 1 is not hyperbolic for any degree d, provided n 2. In fact, such a manifold contains a rational curve:

(u, /0 eP1 (C) —) (u, y, w u, w y, 0, ... , 0)EP„ + ,(C),

where w denotes a d-th root of —1. On the other hand, I know of no example of a compact hyperbolic manifold whose first Chem class is not negative.

If M is of the form DIE', where D is a bounded domain in C" and I' is a properly discontinuous group of holomorphic transformations acting freely on D, then M is hyperbolic and also c i (M) is negative. For the proof of the first assertion, see Kobayashi [9,10]. Let 2 E goc„ de d? be the Bergman metric of D. Since it is invariant by I', it may be considered

also as a metric on M =DIT. Similarly, the 2-form —L E gŒ, .de A d? 2ni

may be considered as a 2-form on M. From the definition of the Bergman metric, it is clear that this 2-form represents the first Chem class ci (M). Since (go) is positive definite, c 1 (M) is negative, thus proving the second assertion. The holomorphic transformation group 5(M) of M =DIT is therefore fmite either by Theorem 2.1 or by Theorem 2.2. The finiteness of 5(M) for M =DIE has been proved by Bochner [2], Hawley [1], Sampson [1 ].

In connection with Theorem 2.1 and one of the examples above, we mention the following result.

Theorem 2.3. Let M be a non-singular hypersurface of degree d in + 1 (C). If n 2 and d 3, then the group 5(M) of holomorphic trans-

formations of M is finite, except in the case when n=2, d= 4.

See Matsumura and Monsky [1 ], where a completely algebraic proof is given. Lemma 14.2 in Kodaira-Spencer [1] shows also that dim $3(M)=0 if n 2 and d. 3. Matsumura and Monsky show that 5(M) can be an infinite discrete group when n =2 and d= 4.

The reader will find also a completely algebraic proof of Theorem 2.1 in Matsumura [1 ].

We say that an algebraic manifold M of dimension n is of general type if

1 sup lim —„ dim H° (M,Km)>O,

m. + co tn-


where K denotes the canonical line bundle of M. The following theorem generalizes Theorem 2.1.

Theorem 2.4. If M is a projective algebraic manifold of general type, then its group 6(M) of holomorphic transformations is finite.

For a completely algebraic proof of this theorem, see Matsumura [1 ]. A transcendental proof can be also given along the same line as the proof of Theorem 2.1. We again map 6 (M) onto a group of linear transfor-mations of the vector space H=1/° (M, ICP) leaving a certain star-like bounded domain D invariant. The only nontrivial part of the proof is to show that this representation is faithful if p is large. But this follows from the result of Kodaira to the effect that we can obtain a projective imbedding of M using a certain subspace of H = H° (M, Kt') for p large. (For the detail, we refer the reader to the Addendum in Kobayashi-Ochiai [2].)

For a compact Riemann surface we have the following very precise result of Hurwitz [1 ].

Theorem 2.5. Let M be a compact Riemann surface of genus p 2. Then the order of the group of holomorphic transformations of M is at most 84(p - 1).

We shall only indicate an outline of the proof. Let V be a compact Riemann surface of genus p' and f: M —0/ an n-fold covering projection with branch points. Let aeM be a branch point. With respect to a local coordinate system z with origin at a and a local coordinate system w with origin at f (a), the mapping f is given locally by w=2" around a. Then m— 1 is called the degree of ramification off at a. Let al , ... , ak be the branch points of f with degrees of ramification ml , ..., m k . Then the Riemann-Hurwitz relation states

k

i.1

where x (M) and x (V) denote the Euler numbers of M and V. This formula can be easily verified by taking a triangulation of V such that f (a 1), . . . are vertices and the induced triangulation of M and then by counting the numbers of vertices, edges and faces.

Let eo be a finite group of holomorphic transformations of M; we know already that the group of holomorphic transformations of M is finite if the genus p of M is greater than 1. Let %. denote the isotropy subgroup of 45 at aEM. Let f: M —) MA5 be the natural projection. If the order m of 05a is greater than 1 and if z is a local coordinate system around a eM, then we introduce a local coordinate system w around


f(a)e114/0 by z= wm. In this way, MAD becomes a compact Riemann surface which we shall denote by V. Then M is a branched covering of V with projection f, to which we apply the Riemann-Hurwitz relation. The degree of ramification of f at a is equal to m-1. If we denote by n the order of 15, then the s-orbit through a consists of n/m points. The sum of the degrees of ramification of f at these points on the s-orbit

1 of a is therefore equal to rTi (m —1) = n (1 -—

m . Hence, the Riemann-

Hurwitz relation is of the form

X(M)+n > (1--1

) =n • X(V)• mi

Since m i is the order of a subgroup of 6, mi divides n. If we denote by p' the genus of V, then the formula above may be rewritten as follows:

2p — 2 =2p' 2+ E (1 1 ).

1=1' mi

If 2, then (p —1)/n1. If p' =1, then (2p— 2)/n> (1 —

and hence n 4(p —1). Finally, consider the case p' =O. Then

k 1 (2p— 2)/n= — 2+ E (1--1 )=k- 2— E

mi

It follows that k3. If k5, then (2p — 2)/n 2 and n 2 2

For k =4, we have the following possibilities:

m2 m3 7774 (2p-2)/n

>2 =2 =2

=2

>2 >2 =2 =2

>2 >2 >2 =2

>2>2>2>2

> 2 = 3.

>1.. = 2

> 1 =T =>1.

ri ir 71. n

3(p— I)

4(p-1) 6(p-1)

12(p— 1)

For k= 3, we have the following possibilities:

.m2 7713 (2p-2)/n

>3 >3 >3 >+ 8(p-1) =3 >3 >3 > 1 =w n12(p-1) .3 =3 >3 =12 n24(p-1) =2 >4 >4 1 n -20(p — 1) =2 =4 >4 72-40(p — 1) =2 =3 >3 n 84(p — 1)

1 mi = 2


Let M be a compact complex manifold and K be its canonical line bundle. Let k be a positive integer such that K k is very ample over some nonempty open set U of M in the following sense. At each point x of M, let H(x) be the subspace of H = (M; K I ) consisting of holomorphic sections of K i‘ vanishing at x. Assume that, for each x e U, H(x) is a hyperplane of H and that the mapping x e U —) H(x) gives an imbedding of U into the projective space P(H*) (see the proof of Theorem 2.1). Then the natural representation p of 5(M) on H is faithful. In fact, if a holomorphic transformation of M leaves every holomorphic section of Ki` fixed, then it leaves every point of U fixed and, being holomorphic, it leaves every point of M fixed. In particular, if M is a compact Riemann surface of genus p 2, then K is very ample over some nonempty open subset U, and it follows that a holomorphic transformation of M leaving every holomorphic 1-form fixed is the identity transformation. Thus we have the following result of Hurwitz DI

Theorem 2.6. A holomorphic transformation of a compact Riemann surface of genus p 2 is the identity transformation if it induces the identity transformation on the first homology group Hl (M ; R).

For higher dimensional analogs of Theorem 2.6, see Theorem 4.4 of Chapter II and Borel-Narasimhan [1].

For more results on automorphisms of compact Riemann surfaces of genus 2, see Macbeath [1], Lehner-Newman [1 ], Accola [1, 2], Lewittes [1 ].

For an analog of Theorem 2.5 for algebraic surfaces, see Andreotti [1]. Somewhat related with the results of this section is the following

theorem of Gottschling [I]. Let H. be the Siegel upper-half space of degree m, i.e., the space of complex symmetric matrices of degree m with positive definite imaginary part and 11 be the Siegel modular group of degree m. Then, for m 3, the group of holomorphic trans-formations of H,./F,, consists of the identity element only.

3. Holomorphic Vector Fields and Holomorphic 1-Forms

If Z is a holomorphic vector field and co is a holomorphic 1-form on a complex manifold M, then w(Z) is a holomorphic function on M. If M is compact, this function must be constant. This simple fact yields some useful results.

Proposition 3.1. Let M be a compact complex manifold and b =1)(M) the Lie algebra of holomorphic vector fields on M. Denote by W i' ° the space of closed holomorphic 1-forms on M. Define

(Zeb; co(Z)=0 for all co EV.°)

3. Holomorphic Vector Fields and Holomorphic 1-Forms 91

Then (1) ih is an ideal of lj and contains [f),1)], (2) dim bib, bi ,

where 1;1 is the first Betti number of M.

Proof We recall the following general formula relating a 1-form w and vector fields Z and W:

2 dco (Z, W) = Z(o)(W))— W(co (Z)) ([Z, W]).

(1) follows immediately from this formula. Let be the space of closed anti-holomorphic 1-forms.

It is a simple matter to verify that V. ° (resp. W°. 1) is the space of closed (1, 0)-forms (resp. (0, 1)-forms). Let V be the space of closed (complex) 1-forms. Then

Let 21 be the space of cobounding 1-forms, i.e., 1-forms of the type d f. Then g. Let

01 . ° = {coeV. ° ; co(Z)=0-for all Zeb},

0°. 1 = fro ee. 1 ; 45 (2-) = 0 for all Z eb) =P,° .

Then the pairing (a), z) ew1,0 x 9 —> co(Z)eC

induces a dual pairing between V.°/01. ° and filth. We shall show

0 ± %90,1) Œ g1,0 ± g0,1 .

Let d f = a+ #effl1 n w1,0 +w0,1• ) where a, fieV. °. Then

df+df=a+fi+d+fi. Let Zelj. Then

(df + df)(Z)=(a+ fi)(Z).

Since a and fi are holomorphic, the right hand side is constant. On the other hand, the left hand side vanishes at the maximum point of the real valued function f + f Hence,

(Œ+ f3)(Z)=0. Similarly, from

(df —df)(Z)= (a — /3)(Z), we obtain (a— f3)(Z)=0.

Hence, a(Z)=13(Z)= 0. This proves our assertion that df is in 01 . ° gO, I.


We may now conclude

2. dim fillyi =2 . dim V. °/gla °= dim (WL ° + W°. 1)/(gl . ° + 0°' 1) dim(" o wo, n o

dim V/R1 = . q. e. d.

Remark. A holomorphic vector field Z with non-empty zero set belongs to the ideal bp

The ideal 1), of 1)(M) was introduced by Lichnerowicz [3] to study b(M) of a compact Kdhler manifold M with non-positive, non-negative or zero first Chern class. See also Matsushima [5].

4. Holomorphic Vector Fields on 'Uhler Manifolds

Let M be a Kdhler manifold of dimension n. Let Z be a complex vector field with components COE, C' in terms of a local coordinate system 2 , z", i.e.,

VZ=V 1 Z+V"Z,

where V 1 Z and V"Z are defined by the property that

ViarZ =0 and V41, Z =0 for all vectors W of type (1, 0).

In terms of a local coordinate system,

a a viz=Evft ccedzPo Ta-z—a +Evpcadzfico

a a v"z= vp a? Ta-zr, +E di' a? •

Similarly, for any tensor field K, we can write

VK=V1 K+V"K.

Given a complex vector field Z of type (1, 0), we denote the (0, 1)-form corresponding to Z by C. In terms of their components, we have

a Z = E cce 4—* c=E Cp d? with Co = E gap Ca.

Proposition 4.1. A complex vector field Z of type (1,0) on a Ktihler manifold M is holomorphic if and only if V"Z =0, or equivalently, V" C=0 (where is the (0, 1)-form corresponding to Z).

We can write

4. Holomorphic Vector Fields on Kahler Manifolds 93

Proof. Clearly, Z is holomorphic if and only if, for each point p of M, there exists a local coordinate system z', zn around p such that (0Cyail) p = 0. On the other hand, since M is a Kahler manifold, for each point p there exists a local coordinate system z', around p such that (V ) p .(alag) (i.e., such that the Christoffel symbols vanish at p). Hence, Z is holomorphic if and only if Vp COE= 0. q. e. d.

Theorem 4.2. Let M be a Kahler manifold, Z a complex vector field of type (1,0) and the corresponding (0, 1)-form. If Z is holomorphic, then

PC=z1"C=ER ŒA CŒdZfi.

Conversely, if M is compact and

g(z1"C—E R ŒA CŒde,C)dv=0,

then Z is holomorphic.

Proof. In Appendix 3, the following general formula for a (0, 1)-form is proved:

A 11 = E(—va vecco nfl+Rap ca de). Since

VŒ= gjA Vy

Proposition 4.1 implies V' CA =0 if X is holomorphic. Hence, we obtain the first statement of the theorem. To prove the converse, we use the following integral formula (see Theorem 4 of Appendix 2) expressed in terms of a local coordinate system:

— " c)# cfl + Rap. ca cfl + »cce • VTG} d v 0 .

The first two terms of the integrand cancel each other by our assumption. Hence,

Cc • VIZ-, d v = O.

This implies » COE = 0. By Proposition 4.1, Z is holomorphic. q. e. d.

The first half of the theorem is due to Bochner [1] and its converse to Yano [4].

Theorem 4.3. Let M be a compact Kahler manifold and Z= X —iJX a complex vector field of type (1, 0) with real part X. Then X is an infini-tesimal isometry if and only if Z is holomorphic and div X =0.

Proof. We use the characterization for an infinitesimal isometry obtained in Theorem 2.3 of Chapter II. By Theorem 4.2 Z is holomorphic if and only if X satisfies (1) in Theorem 2.3 of Chapter II. q. e.d.


Theorem 4.3 is due to Yano [4].

Theorem 4.4. Let M be a compact Kahler manifold and Z a holomorphic vector field (of type (1, 0)) with the corresponding (0, 1)-form C. Then

(1) C=H C + d" f, where H1 is the harmonic part of and f is a ( complex valued) function. Such a function f is unique if it is normalized by the property fdv=0.

(2) (=d" f if and only if œ(Z)= 0 for every holomorphic 1-form a, i.e., if and only if Z e fh, where b l is the ideal introduced in § 3.

(3) The real part X of Z is an infinitesimal isometry if and only if the real part off is a constant. (This means that if f is normalized as in (1), then f is purely imaginary.)

Proof By Proposition 4.1, d" (=O. Now, (1) follows from the Hodge-Kodaira decomposition theorem:

C=HC-Fd"6"q)+6"d"(p,

where q) is a form with the same bidegree as C. If C = H C + d" f = H C + d" g, then d"(f — g)= 0, that is, f— g is holomorphic and hence a constant.

To prove (2), we observe first that if a is a holomorphic 1-form, then Œ(Z) is a holomorphic function on M and hence is a constant. It suffices therefore to prove that the integral j. a(Z) dv vanishes for all holomorphic

1-forms a if and only if H C = O. Assume H C= 0. Then a (Z)= g (a, C)= g(Œ, d"f), where g denotes the inner product on the cotangent spaces defined by the metric. Since a is holomorphic, it is harmonic. Since a harmonic form is perpendicular to d'f=d"f in the Hodge-Kodaira decomposition, the integral g(a, d" f)dv vanishes. Assume, conversely,

that œ(Z)= O and let a= H C; since H? is a harmonic (0, 1)-form, its complex conjugate is a harmonic (1, 0)-form and hence is holomorphic. Then

0= 11((Z)dv= J (g(HC,H C)+g(II C,d"f))dv= J g(II C,H ()dv.

Hence, H=0. X is an infinitesimal isometry if and only if div X =0 (Theorem 4.3).

On the other hand, div X =0 if and only if 6(C +Z) =0. But

= (ô" d" + d" (5") f + (ô' d' +d' 6') f =d" f + f =Of +Itlf=1,61(f+f).

5. Compact Einstein-KOhler Manifolds 95

Since d(f + J7)= 0 if and only if f + f is a constant, we obtain the assertion in (3). q. e. d.

(I) and (2) of Theorem 4.4 are due to Lichnerowicz [6].

Corollary 4.5. In Theorem 4.4, if the zero set of Z is nonempty, then 1=d" f.

Proof. If a is a holomorphic 1-form, then œ(Z) is constant. If the zero set of Z is nonempty, then a (Z)= O. Our assertion follows from (2) of Theorem 4.4. q. e. d.

Corollary 4.6. In Theorem 4.4, assume that the real part X of Z is an infinitesimal isometry. Let C be the (0, 1)-form corresponding to Z and be the real 1-form corresponding to X. Then the following statements are mutually equivalent:

(1) The zero set of Z ( =the zero set of X) is nonempty; (2) C = d" f, where f is a function with purely imaginary values; (3) c =Jdu, where u is a real valued function.

5. Compact Einstein-KHhler Manifolds

In this section we shall prove the following result of Matsushima [I].

Theorem 5.1. Let M be a compact Einstein-Kahler manifold with non-zero Ricci tensor. Then the Lie algebra i(M) of infinitesimal isometries is a real form of the Lie algebra 14(M) of holomorphic vector fields, i.e.,

1)(M)=i(M)+{:i • t(M).

In the statement above, i(M) is imbedded in t(M) by identifying an infinitesimal isometry X with the corresponding holomorphic vector field Z=X—iJX (see Theorem 4.3).

Proof. By our assumption, the Ricci tensor RŒ and the metric tensor &co satisfy the relation:

Ro=c • go, where c is a nonzero constant.

Let Z be a holomorphic vector field (of type (1, 0)) and C the corresponding (0, 1)-form. By Theorem 4.2,

A" C = c C .

Substituting C = H + d" f (see Theorem 4.4, (1)) into this, we obtain

A" d" f=c(HC+ d" f),


which shows that El( =O. We may assume that f is normalized as in Theorem 4.4 and we write

f=u+i v,

where u and v are real valued functions. We shall show that

.4" d"u=cd"u and LI" d"v=cd"v.

Since ( =d"f and d" C = c C, we have

0= d"d"f — cd"f=d"4"f — d" cf=d"(A" f—cf),

which shows that d"f —cf is a holomorphic function and hence is a constant. But

d" f —cf =(A"u—cu)+/(61" v—cv)=(4z1u—cu)+4.1z1v—cv).

Hence, both the real part (z1"u— c u) and the imaginary part (X v— c v) of d"f —cf must be also constant. It follows that d"(z1"u — c u)= 0 and d"(z1" v— c v)=0, showing our assertion.

By Theorem 4.2 this means that the vector fields U and V of type (1,0) corresponding to the (0, 1)-forms du" and i d" v are holomorphic. By (3) of Theorem 4.4, —iU and V correspond to infmitesimal isometrics since —iu and iv are purely imaginary. Since C = d" f = d" u + i d" v, we have

Z= i ( —i U)+ V.

This shows that b ( w). i(m)+1/=-T. • i(M). q.e. d.

In the course of the proof, we established that d"f — cf is a constant. Integrating d"f —cf over M and observing that the integral of z1"f (=id f) vanishes, we see that this constant is zero if f is normalized. Hence, ef=cf, or

zlf . = 2 c f.

Denote by "2 , the set of all complex valued functions f which are eigen functions of the Laplacian LI with eigen value 2c, i. e.,

Ac= {f ; df= 2cf} .

Then the correspondence f—C=d"f—)Z

gives a complex linear isomorphism between g$72 , and b(M). The subspace of F2 , consisting of purely imaginary functions corresponds to i(M).

Since b (M)= 0 if c < 0 by Theorem 2.1, Theorem 5.1 is of interest only when c> 0.

6. Compact Kdhler Manifolds with Constant Scalar Curvature 97

Theorem 5.2. Let M be a compact Kahler manifold with vanishing Ricci tensor. Then the Lie algebra 1)(M) of holomorphic vector fields coincides with the Lie algebra i(M) of infinitesimal isometries. It consists of parallel vector fields and is abelian.

Proof. Let Z be a holomorphic vector field and c the corresponding (0, 1)-form. By Theorem 4.2,

that is, is harmonic. In the decomposition =1-1C+d"f in (1) of Theorem 4.4, the function f is zero (if it is normalized). By (3) of Theorem 4.4, the real part of Z is an infinitesimal isometry. This estab-lishes 13(M)= i(M). By Corollary 4.2 of Chapter II, every infinitesimal isometry of M is a parallel vector field. Clearly, every parallel vector field is an infinitesimal isometry. Finally, the general formula

[X, Y] —(Vx Y — Vy X) = 0

implies that the Lie algebra of parallel vector fields is abelian. q. e. d.

For a generalization of Matsushima's theorem to compact almost Einstein-Kahler manifolds, see Sawaki

It is interesting to find out how large the class of compact Einstein-Kahler manifolds is. It is not known if there exists a non-homogeneous compact Einstein manifold. In this connection, see Berger [2] for a survey on Einstein manifolds and Aubin [1] for a construction of certain Einstein-Kahler metrics.

6. Compact Kkhler Manifolds with Constant Scalar Curvature

In this section we shall prove a theorem of Lichnerowicz [2, 3] which generalizes Theorem 5.1 of Matsushima.

Theorem 6.1. Let M be a compact Kahler manifold with constant scalar curvature. Let b(M) denote the Lie algebra of holomorphic vector fields, i(M) the Lie algebra of infinitesimal isometries (considered as a sub-algebra of 1)(M)), c the subalgebra of b(M) consisting of parallel holo-morphic vector fields and b the ideal of b(M) consisting of those vector fields Z such that a(Z)=0 for all holomorphic 1-forms Œ. Then

(1) 13(M) =b + e ( Lie algebra direct sum);

(2) b = (1(M) r b) + (M) b), i.e., i(M)n b is a real form of b, (3) i(M) = (i(M) n b) + c.

Proof. We denote by Q the linear endomorphism on the space of (0, 1)-forms defined by

QC=E RŒff COE dg for


Let Z be an arbitrary holomorphic vector field on M and C the corre-sponding (0, 1)-form. Following Theorem 4.4, we write

C= 9 + d"f, where 9= H1 =the harmonic part of C.

As in § 5 we write

f = u+ iv, where u and y are real valued functions. We set

= d" u and ti = id" v so that

We shall show that 9 corresponds to a vector field belonging to c and that c and q correspond to vector fields belonging to {:i(i(M)r) b) and i(M) r) b, respectively.

We shall make use of the following formula which follows easily from the second Bianchi identity:

Lemma. If we denote by R the scalar curvature, then

EveRap=vp-R and EvoR=va R.

Since 9 is a harmonic form of degree (0, 1), its conjugate is holomorphic and hence V1, 941 = O. This fact, together with Lemma, implies

d" (Q 9)= —Evo(R„, (pa)=—E 9Œ Da R=0.

Since Z is holomorphic, Theorem 4.2 implies

d"C—QC=0. Since d" 9= 0, this implies

f)— Q(d" f)= Q q. Hence,

S (d"(d" f)— Q(d" f), d" f) dv= 5 (Q9, d" f) dv

= 1 (5" (Q 9), f) dv= O.

By Theorem 4.2, this means that the vector field corresponding to the (0, 1)-form d" f is holomorphic. Hence, the vector field corresponding to 9 (=C— d" f) is also holomorphic. By Proposition 4.1, Vii 941 = O. Since we already know that Vo 941 =0, we can conclude that 9 is parallel.

Since d" f = + n corresponds to a holomorphic vector field, Theorem 4.2 implies d" — Q = — (d" ri — Qt». Hence

6" (d" — Q ) = - (5" (A" q — Q q) -

6. Compact KKhler Manifolds with Constant Scalar Curvature 99

We shall show that both sides of this equality vanish by demonstrating that the left hand side is real and the right hand side is purely imaginary. We have

Q )= (5"d" 6" — (5" (Q

= d" (5" d" u+ E vo(RŒA • Vau)

=.4" • tru+ER„ A •VP- Vau.

This shows that 6" (A" c — Q 0 is real. Similarly,

(5" (A" — Qq). i(z1" A" v + ROE, • vo va v),

which shows that 6" (A" — Qq) is purely imaginary. Hence,

Now we have

(.4" dv= j(z1"—N,d"u)dv

=

By Theorem 4.2, corresponds to a holomorphic vector field. Hence, corresponds also to a holomorphic vector field.

Denote by Zo , Zi , Z2 the holomorphic vector fields corresponding to the (0, 1)-forms cp, respectively. Then

Z = Zo + + Z2 .

We have shown already that Zo is in c. By (2) and (3) of Theorem 4.4, Z1 is in {:1(i(M) r b) and Z2 is in i(M) n b. The facts that

b n c= 0 and (i(M) n n (i(M) b)) = 0

are also immediate consequences of Theorem 4.4. To prove that [b, =05 let

a a z= Vœf

aza e b and W= (pa

azŒ e c.

Then

a a

Ez,141= (vaf. Vcc (PP — 49P VI3V œf) = — E koft VP Vœf)

a

=— (9P vavp f) 2- - E \Ng" vp f) aza aza


since W is parallel. But

E9-73v,I=E0,1v.f. Since E 9A dg is a harmonic (0, 1)-form, Eço,dzP is a harmonic (1, 0)- form and hence is a holomorphic 1-form. We set oc= dzP. On the

a other hand, the vector field Z' =E vPi is also in b. Since ot(Z)=

azP by Theorem 4.4, we obtain

v" f.o.

This completes the proof of the fact that [Z,W]= 0. Finally, (3) follows from (1) and the fact that every element of c, being

parallel, is in i(M). q. e.d.

7. Conformal Changes of the Laplacian

In order to study compact Kahler manifolds with non-negative or non-positive first Chan class, it is convenient to introduce the Laplacian with respect to a hermitian metric conformal to the given Kahler metric. The results in this section are due to Lichnerowicz [6]. We follow both Lichnerowicz [6] and Matsushima [5].

Let M be a Kdhler manifold and let e be a real, positive function on M. We introduce operators 45; and A operating on complex differential forms by

(5:', = e a" (ea .0 for every differential form ço, and

A :1,= o + d" o 5:.

By a direct calculation using local coordinates (see Appendix 3), we obtain

(51;9=6"9-1(dio-)9,

where t(di a) denotes the interior product of 9 with the complex vector field of type (0, 1) corresponding to the (1, 0)-form d'a. We obtain easily

4.= — (d" 0 t(d' a)-F t(d 1 a) 0 d").

If M is compact, we define a new inner product ( , )a by

(9, Oa = S (9, tTi) • e • dv

Then (d" (P) cT&P) 13';

7. Conformal Changes of the Laplacian 101

In fact,

(d" 9, ky = (d" 9, e tT)= (9, 6" (e til)) = (9, e'' 6" (e 111))0 = (9,

A differential form 9 is said to be 4-harmonic if di:, 9=0.

Theorem 7.1. Let M be a compact Keihler manifold and denote by the space of 4-harmonic (p, q)-forms and by OP the sheaf of germs

of holomorphic p-forms over M. Then

(1) dim H:.q< oo;

(2)

In particular, dim He" is independent of a.

The result (1) is due to Kodaira and (2) is the result of Dolbeault. For the proof and further references, see Hirzebruch [1; Chapter IV].

We define a tensor field with components Co by

Co = Rap — VilVa a,

and denote by Q a the linear endomorphism on the space of (0, 1)-forms defined by

Q a C = E C (HP for C= E Ca dr .

Then the following theorem generalizes Theorem 4.2. Theorem 7.2. Let M be a Kiihler manifold, Z a complex vector field

of type (1, 0) and C the corresponding (0, 1)-form. If Z is holomorphic, then

4C—Q a (C)=0.

Conversely, if M is compact and

(A 1(11C — 02„(0, = 0 '

then Z is holomorphic.

Proof We prove the following lemma which is a local formula.

Lemma. A — Q a (C)= d"C — Q(C)—Vs C, where Q= Q 0 and S is the vector field of type (0, 1) corresponding to the (1, 0)-form d'a.

a Proof ofLemma. Since d' a =EV„a dza and S=EVa Ta--F, we obtain

d" 0 t(d 1 a) = d"(E Vi a. CO= E (v„ vet c„ + vet a • VA CO dg,

t (d' a) 0 d" = E Va cr (Va Cp— Vp CO dg.


•

Hence, 461::C=A"C—(d" 0 t(d' a) - F t(d' cr) 0 d") C

= A" C — E eir a - VŒ C0 + V 0V a • ca) (1513

= A" C—Vs C—EVA Va a -CŒ difl .

Now the lemma follows from the following fact:

Suppose Z is holomorphic. Since S is of type (0, 1), Proposition 4.1 implies Vs ( =0. Then by Lemma,

The first half of Theorem 7.2 follows from Theorem 4.2. To prove the second half, we make use of the following integral

formula (see Appendix 2):

(A " C — Q(O, 5) = (v" C, V" co),

where C and co are arbitrary (0, *forms. We set w= e c. Then the left hand side is equal to (A" C — Q (0, 0, while the right hand side is equal to

1 (E VA C' V(e. Ca)) dT/= 1 (E VA CŒ • VA a • Ca + E vp-cce • Vif C Œ) e • d1/

=( , VsC)k+ (V" CV" CL. Hence,

(A" C — Q(0 — v s C, 0 , = (v" c,v" 0, .

By Lemma, this formula may be written as

(A icir C — Qff (0) 0, = (V"

Now, if the left hand side vanishes, then V"C = 0. By Proposition 4.1, Z is holomorphic. q. e. d.

Let dv denote the volume element of a Uhler manifold M and define a positive 2n-form Q= e dv. Let bc, be the Lie algebra of holomorphic vector fields leaving the form 0 invariant, i.e.,

bc,= (z eb(M); Lz fl = 0) .

Proposition 7.3. For a compact lathier manifold M, the subalgebra k of the Lie algebra b(M) of holomorphic vector fields may be defined by

= (Z et)(M); 44 C=0) ,

where C denotes the (0, 1)-form corresponding to Z.

8. Compact Kahler Manifolds with Nonpositive First Chern Class 103

Proof We have

Lz 0=Lz (ea)dv+e' • L z dv=L z a - 0—ea - 6"C • dv

= (I (d'a) C — 6" C) fl = — 6:1,C - O.

This shows that LO= O if and only if = O. By Proposition 4.1, d" ( = 0 whenever Z is holomorphic. Hence,

for an element Z of 1)(M), 6'; ( = 0 is equivalent to 6'; C= d" C = O. This in turn is equivalent to d; C=0. q.e. d.

Theorem 7.4. Let M be a compact Kahler manifold. Then the sub-algebra t) 1)(M) possesses the following properties:

(1) b , is abelian; (2) if Zell, and co(Z)=0 for all holomorphic 1-forms co, then Z=0

(that is, ber n th = 0, where th is the ideal of 1)(M) introduced in § 3);

(3) dim 1)0. .1- b 1 , where b 1 is the first Betti number of M; (4) if Z is a nonzero element of lx„ then the zero set of Z is empty.

Proof We recall that 1) 1 is the ideal of 1)(M) consisting of vector fields Z such that w(Z)= O for all holomorphic 1-forms co. (Since M is a Kdhler manifold, every holomorphic form is closed.) Since 1) 1 contains the derived algebra of t) (M) (see Proposition 3.1) and since f) , is a subalgebra, we have

[IL , IL] OE IL n 1)1 .

Let Zel),,,n bi• By (2) of Theorem 4.4, we have C=d"f. Hence,

(C, Oa = fri"f, 4, = (i; 6'; Oa = ° by Proposition 7.3. Therefore, Z= 0, thus proving (2) and hence (1). Now (3) follows from Proposition 3.1 and the inclusion

IL = ILAI), n 1)1) c= b(m)/bi .

Suppose Zek and Zero(Z)#Ø. As we remarked at the end of § 3, Z must be in 1) 1 . By (2), Z = O. This proves (4). q.e. d.

8. Compact Khler Manifolds with Nonpositive First Chern Class

Let M be a compact complex manifold and c1 (M) its first Chern class. We say that c 1 (M) is nonpositive and write ci (M) 0 if it can be rep-resented by a closed (1, 1)-form

i y = 2i, E Co de A d9


such that (Co) is negative semi-de finite hermitian. We have already considered manifolds with negative first Chern class (see §2).

Assume that M is a compact KMler manifold with Ricci tensor R Then c1 (M) is represented by

p=--- E RŒ dZŒA d.V. 27r

By Theorem 1 of Appendix 4, a (1, 1)-form p is cohomologous to y if and only if there exists a real valued function a such that

Cao = Rap — VOE a.

We prove two theorems of Lichnerowicz [6] (see also Matsushima [5]).

Theorem 8.1. Let M be a compact lahler manifold with c1 (M).0 and t) (M) be the Lie algebra of holomorphic vector fields on M. Then

(1) b(M) is abelian and dimb(M)-1-13 1 . (2) If Z is a nonzero element of t) (M), it never vanishes on M.

(3) If a closed (1, 1)-form y =-2

in E Cap dza A dZI3 represents ci (M) and

(Ce) is negative semi-definite everywhere and negative definite some-where, then 1)(M)= 0.

Proof Let Z be a complex vector field of type (1,0) with the corre-sponding (0, 1)-form C. By Theorem 7.2, Z is holomorphic if and only if

(z1 :1,C — Q7(c), =0. This is equivalent to

c„„ cec v= 0.

Since (Co) is negative semi-definite, this equation is equivalent to

d"C=0, (5 .C=0, CŒp COE = O.

In other words, Z is holomorphic if and only if C is i1-harmonic and satisfies E COEfi ca = 0. By Theorem 7.1,

dim f)(M)_._ dim H° . 1 (M ; C).

If (Cap) is negative definite at a point p and Z is holomorphic, then C must vanish in a neighborhood of p and hence everywhere on M.

Let 1)1 be the ideal of f)(M) consisting of holomorphic vector fields Z such that a(Z)=0 for all holomorphic 1-forms Œ. In general, 1)(M)/f) 1 is abelian (Proposition 3.1). We shall show that f 1 =0. Let Z efh. By

8. Compact Kghler Manifolds with Nonpositive First Chan Class 105

Theorem 4.4, C=d"f, where f is a function. On the other hand, C is harmonic. Hence, C=0.

Since '4, =0, 13 (M) itself is abelian. Suppose the zero set of Z is non-empty. If Z is holomorphic, for every holomorphic 1-form a the function Œ(Z) is holomorphic and hence constant. Since a(Z) vanishes at some point, it must vanish identically. This means that Z is an element of 13 1. Hence, Z=0. q. e. d.

Corollary 8.2. Let M be a compact Kahler manifold with c 1 (M) 0. If r is the maximal rank of (Cap), then dim1)(M)n—r, where n is the complex dimension of M.

Remark Theorem 8.1 implies that if cl (M)<O, then 13 (M)=0. But this is also a direct consequence of the vanishing theorem of Kodaira [1]- Nakano [1]. In fact, if we denote by SP the sheaf of germs of holomorphic p-forms and by T and T* the bundles of complex vectors of type (1,0) and (1, 0)-forms respectively, then

b (ti) Ho ; f2 ° (I)) dzl Hn(M ; On (T*))= lin(M ; (K)),

where K is the canonical line bundle of M. If ci (M)<O, i.e., ci (K)>O, then the Vanishing theorem implies Hn(M; 01 (10)=0. But in Theo-rem 2.1, we proved a stronger statement that if ci (M)<O, then the group b(M) of holomorphic transformations of M is-finite.

Theorem 8.3. Let M be a compact Kahler manifold with c 1 (M)=0. Let 13 (M) be the Lie algebra of holomorphic vector fields of M. Let V. °

denote the space of holomorphic 1-forms on M. Then

(1) 13 (M) is abelian and dimb(M)=ib i , where b 1 is the first Betti number of M.

(2) The ideal th= {Z E (m); co (z) =0 for all w e' CI ) is trivial. In particular, for every nonzero Zel)(M), its zero set is empty.

-A., (3) The bilinear mapping (co, Z)ew xb(M)—)a)(Z)EC is a dual pairing between W ." and b(M).

Proof We prove the following lemma first.

Lemma. Let M be a compact Kahler manifold with c 1 (M) 0. Choose a function a in such a way that (Cap)_ 0 (see § 7). Let H" be the space of 4-harmonic (0, 1)-forms and let bc be the ideal of 13 (M) defined by

= (Zeb(M); d; C=0) (see Proposition 7.3). Then Z —)C gives an iso-morphism from b„ onto H"..

Proof of Lemma. Let Cell'''. We make use of the following formula established in the course of the proof of Theorem 7.2:

(A i' C — C7(0, = (v" C, V" Oa


Since z1,7"C ,---0 and (C,0)0, the left hand side is non-positive. Hence, V"C=0. By Proposition 4.1, the corresponding vector field Z is holo-morphic. This proves that her —) H" is surjective and hence is bijective. This completes the proof of Lemma,

Lemma implies that if c 1 (M)_0, then

dim ba =---ibl . If c 1 (M)=0, then Theorem 8.1 implies that 13 (M) is abelian and

dim 13(M) -61 bi .

dim b(M)=Ibi .

This proves (1). Since 1)(M)=1)„, (2) of Theorem 7.4 implies 13 1 =-0. This proves (2). We know (see § 3) that the bilinear mapping

x

induces a dual pairing between W1 ' °/g1 ' ° and f)(M)/fh (see also § 3 for the definition of gl i 0). Since 13h=0 and dim b(M) = b 1 = dim W1 ' °, it follows that 0=0. Hence, the bilinear mapping above is a dual pairing between V. ° and b(M). q.e.d.

For results which sharpen Theorem 8.3, see Matsushima [4, 6].

9. Projectively Induced HolomOrphic Transformations

Let M be a compact complex manifold and (W) the group of holo-morphic transformations of M. Given a subgroup 0 of b(M), we ask if there exists an imbedding of M into a complex projective space PN (C) such that e• is induced by (i.e., the restriction of) a group of projective linear transformations of PN (C). We begin with the simplest case.

Theorem 9.1. Let M be a compact complex manifold with positive or negative first Chern class c1 (M). Then there is an imbedding of M into a complex projective space PN (C) such that every holomorphic transforma-tion of M is induced by a unique projective linear transformation of PN (C).

Proof The case c1 (M)<0 is contained in the proof of Theorem 2.1. Let K be the canonical line bundle of M. Then K is ample. For a suffi-ciently large integer p, the space H° (M; KP) of holomorphic sectiàns of KP contains sufficiently many sections to induce an imbedding of M into PN (C), where N =dim H° (M; .10-1. Every holomorphic trans-formation of M induces a linear transformation of H° (M; KP) and hence a projective linear transformation of PN (c). See the proof of Theorem 2.1 for the details. If c i (M)>O, then K-1 is ample and a similar argument using H° (M; K - P) with a large integer p proves the theorem. q.e.d.

Hence, 13(m)=13,, and

9. Projective ly Induced Holomorphic Transformations 107

The following theorem is due to Blanchard [1] and Borel.

Theorem 9.2. Let M be a compact Hodge manifold with first Betti number b l = O. Then there is an imbedding of M into a complex projective space 4, (C) such that every element of the largest connected group b °(M) of holomorphic transformations is induced by a unique projective linear transformation of M.

Proof Let Q (resp. Q*) be the sheaf of germs of holomorphic func-tions (resp. holomorphic functions without zero) on M. The exact sequence 0_).z_)g-2 exi__L2r+c• Q* —+1

induces an exact sequence

(M ; 0) —> 111 (M ; 0*)-1-1 2 (M ;Z).

It is essentially by definition that 111- (M; is the group of (isomorphism classes) of complex line bundles over M. By the theorem of Dol-beault, 111 (M ; 0) is isomorphic to the space H" of antiholomorphic (0, 1)-forms, i.e., harmonic (0, 1)-forms on M in a natural manner. Since bl =0, we have H' (M; S2)=0. Hence, H1 (M ; 11 2 (M; Z) is injec-tive. In other words, every complex line bundle of M (i.e., every element of Hl(M; Q*)) is uniquely determined by its characteristic class (i.e., its image in 11 2 (M; Z)). Since b° ( ti) is a connected group, it acts trivially on 112 (M ; Z) and hence also on 111 (M ; 0*). This means that if feb ° (M) and L is a complex line bundle over M, then f* L=L. In other words, f induces an automorphism f of L compatible with f It should be per-haps pointed out that the group ,° (M) may not act on L since fog with f, geb°(M) may not induce fo, that is, we may not have (fig)= Jo k. This is due to the fact that f does not determine f uniquely. If f and f are two automorphisms of L compatible with f, then f(w)= _TM • a(z) for W E L and z=n(w)eM, it denoting the projection L—M. Since a(z) is holomorphic in z and does not vanish, it is a nonzero constant. This shows that although f is not unique, it is unique up to a nonzero constant multiple. Hence, (fig).(10 g') a, where a is a non-zero constant.

Since M is a Hodge manifold, the classical theorem of Kodaira [2] implies that there exists a very ample line bundle L so that H° (M; L) contains sufficiently many sections to induce an imbedding of M into

(C), where N =dim H° (M; L)— 1. Each element f of e(M) induces an automorphism f of L, which in turn induces a linear transformation p (f) of H° (M; L). Again, p may not be a representation in the strict sense but it satisfies the relation

p(fo g).p(f)o p(g)• a,


where a is a nonzero scalar. Hence, considered as a transformation of PN (C), p (f) is uniquely determined byf and p (f o g) = p (f) o p (g). q.e.d.

Remark 1. Actually we have shown in Theorem 9.2 that for any imbedding of M in PN (C) the group b° (M) can be realized as a trans-formation group of PN (C). If we choose an imbedding of M into PN (C) more carefully, the group b° (M) can be given by a linear group acting on CN-1-1 H° (M; L)), that is, p can be made into a representation of b0 (iii) on H° (M; L) in the strict sense. This follows from the following general result.

Proposition 9.3. There is an imbedding of PN (C) into some Pm (C) such that the projective linear group PGL(N; C) is represented by linear trans-formations of the corresponding vector space Cm+ 1 .

Proof Let V be the space of symmetric tensors over CN +' of degree N +1, i.e., the space of homogeneous polynomials of degree n+ 1 over the dual space of CN + 1 . We consider PGL(N; C) as the image of SL(N +1; C) acting on 0+ 1 . Then the center {a 11.1 +1; aN+1 = 1 } of SL(N+ 1; C) is precisely the set of elements which induce the trivial transformation of PN (C). The group SL(N+ 1; C) acts on V in the obvious manner and its center acts trivially on V Hence PGL(N; C) has a faithful representation in GL(V). This is compatible with the obvious imbedding of PN (C) into P„,(C), where m+ 1 =dim V q.e.d.

Remark 2. In Theorem 9.2, if H2 (M; Z)= Z, then not only the identity component .5° (M) but also the whole group b(M) acts trivially on 112 (M; Z) and hence Sj (M) can be realized as a group of projective linear transformations compatible with a projective imbedding of M.

For complex tori, we cannot have results similar to Theorems 9.1 and 9.2. In fact, we prove the following theorem of Blanchard [1 ].

Theorem 9.4. Let M be a Hodge manifold, b(M) the group of holo-morphic transformations and 1)(M) the Lie algebra of holomorphic vector fields of M. As in § 3, let th be the ideal of b(M) consisting of Z eb(M) such that a(Z)=0 for all holomorphic 1-forms Œ. Let bi be the connected normal subgroup of b(M) generated by 1) 1 . Then given a connected Lie subgroup 15 of si (M), there is an imbedding of M into P,,, (C)such that every element of 15 is induced by a unique projective linear transformation of PN (c) if and only if 15 is a subgroup of

The proof is divided into several lemmas.

Lemma 1. Let M be a complex submanifold of a Kahler manifold P. Let Z be a holomorphic vector field on P and the corresponding (0, 1)- form on P If the restriction of Z to M is a (holomorphic) tangent vector field of M, i.e., Z is tangent to M at each point of M, then the restriction of to M is the (0, 1)-form corresponding to the restriction of Z to M.

9. Projective ly Induced Holomorphic Transformations 109

Proof of Lemma 1. Let ( )p and ( , )m denote the inner produéts defined by the Kdhler metric of P and the induced Kdhler metric of M, respectively. Let W be any complex tangent vector of M of type (i, 0). Then

W)m=(Z, W)p= C(W

This proves Lemma 1.

Lemma 2. Let M be a closed complex submanifold of a compact Kahler manifold P. Denote the ideal 1), of b(M) defined in Theorem 9.4 by 1) 1 (M) and denote the similarly defined ideal of b(P) by 1) 1 (P). Let Z be an ele-ment of 1h(P) such that its restriction to M is tangent to M. Then its restriction to M is in 1) 1 (M).

Proof of Lemma 2. Let be the (0, 1)-form corresponding to Z. By Theorem 4.4, C= d" f for some function f. Then Clm = d" (flu). By Theo-rem 4.4 and Lemma 1, Zim is in Ih(M).

We are now in a position to prove a half of Theorem 9.4. Let P be a compact Kdhler manifold without (non-trivial) holomorphic 1-form such as a complex projective space. Then I 1 (P)=1)(P). Then Lemma 2 implies that any holomorphic vector field on M which comes from a holomorphic vector field of P is in the ideal lh(M). This shows that in Theorem 9.4 if 0 comes from projective linear transformations of PN (C) in which M is imbedded, then 0 is contained in

To prove the remaining half of the theorem, it is clear from the proof of Theorem 9.2 that all we have to do is to find a very ample line bundle L over M such that every transformation belonging to bi can be lifted to an automorphism of L. In other words, it suffices to find a very ample line bundle L over M such that every holomorphic vector field of M belonging to lh can be lifted to a holomorphic vector field on L. Actually we prove

Lemma 3. Let M be a Hodge manifold and L an ample line bundle over M. Then every holomorphic vector field of M belonging to k can be lifted to a holomorphic vector field of L.

Proof of Lemma 3. Choosing a hermitian fibre metric h in the line bundle L, let co be its connection form on the associated principal bundle L*. We note that L* can be obtained from L by deleting the zero section of L. Since L is ample, we may assume that h was so chosen that its curvature is positive. We shall write down h, co and the curvature explicitly.

Let U be a small open set in M with local coordinate system z', We fix a holomorphic section a of L* over U. Then we can identify L*I u with U x C*. We denote by t the natural coordinate system in C*. Then we can take zi, e, t as a coordinate system in L*I u = U x C.

110 IlL Automorphisms of Complex Manifolds

Let hu be the square of the length of the section a; it is a positive function on U. We define a (1, 0)-form cou on U by

cou = d' (log he).

Then cou is the pull-back of the connection form co by a, and we have

1 co=cou +—

tdi=d(log h u tt).

The curvature form is given by

d" co u = d" d' (log hu)= E gik dz i A 0', where

gik = — a2 log hula? avc.

We use 2E ga d? (IV` as our Kdhler metric on M. Let

. a

be a holomorphic vector field on M belonging to 1), and the corre-sponding (0, 1)-form. By Theorem 4.4, there is a function f on M such that

=d" f.

We lift Z to a vector field 2 on L* of type (1,0) such that

-f= (0(2).

This determines 2 uniquely. If we write

a a 2=Eci —aii+ T at 7

then

-f= WO = COU(Z)± •

We want to show that 2 is holomorphic, i.e., 't is holomorphic. We have

— d" f= d" (co u (Z))+ d" (r/t)= d" o

Since Z is holomorphic, we have

d" 0 t z + t z 0 d" =O.

9. Projectively Induced Holomorphic Transformations 111

Hence, — d" f = — t z o d" co u + d" (T / t)

= — t z (E g o dzi A de) + d"(r/t)

= —E go ci + d" (TM= —C+ d"(t/t).

Since d"f =C, it follows that d" (TM =O. Since d" t = 0, this implies d" = 0, thus proving that t is holomorphic. We have thus proved that Z can be lifted to a holomorphic vector field 2 on L*. Hence, Z can be also lifted to a holomorphic vector field on the associated bundle L. This completes the proof of Lemma 3. q.e.d.

From Lemma 3 in the proof above, we obtain

Proposition 9.5. Let M and fh be as in Theorem 9.4 and L be a com-plex line bundle over M. Then every vector field on M belonging to bi can be lifted to a holomorphic vector field on L.

Proof We have proved this proposition when L is ample. It suffices therefore to show that the given line bundle L is of the form Li L-2, where both L1 and L2 are ample line bundles. Take any ample line bundle F over M and let k be a large integer such that LFic is ample. Then set L1 = LFk and L2 = 0. q.e.d.

From the definition of th it is clear that a holomorphic vector field Z on M belongs to fh if its zero set is non-empty. It follows that if the Euler number of M is nonzero, then every holomorphic vector field Z belongs to th. More generally (see § 12, Corollary 12.2), if some Chern number of M is nonzero, then every holomorphic vector field Z vanishes at some point and hence belongs to th. Hence

Theorem 9.6. Let M be a Hodge manifold such that some Chern num-ber (e.g., the Euler number) is nonzero. Then M can be imbedded into

PN(C) in such a way that the largest connected group 5° (M) of holo-morphic transformations is induced by projective linear transformations of PN (C )

We shall now prove the following theorem of Matsushima [4].

Theorem 9.7. Let M be a Hodge manifold and 1)1 the Lie algebra of holomorphic vector fields Z such that a(Z)=0 for all holomorphic 1-

forms Œ. Then fh coincides with the set of holomorphic vector fields with non-empty zero set.

Proof We know that a holomorphic vector field Z with non-empty zero set belongs to th because a(Z) is constant for any holomorphic 1-form Œ. Conversely, if Z belongs to th, then Z is induced by a holo-


morphic vector field of a projective space PN (C) in which M is imbedded. It suffices therefore to prove the following

Lemma. Let M be a closed complex subspace in PN (C) and ço, be a 1-parameter group of projective linear transformations of PN (C) leaving M invariant. Then q), has a fixed point in M.

Proof of Lemma. The proof is by induction on N. Lemma is trivially true for N= 1. Assume that Lemma holds for N — 1. Represent ço, as a 1-parameter group of linear transformations of the vector space CN-1-1 . If we choose a suitable basis for CP' -", we can represent c by triangular matrices with zeros below the main diagonal. Let eo , eN be such a basis and 17 be the N-dimensional vector subspace of 0+ 1 spanned by e0 , ..., eN _ l . Then I/ is invariant by (p t . If we denote the projective space associated to V by PN _ 1 (C), then PN _ 1 (C) is a hyperplane in PN (C) invariant by (p,. The hyperplane section PN,(C)nM is a (non-empty) closed complex subspace of PN _ 1 (C) invariant by w By the inductive assumption, q), has a fixed point in PN _ 1 (C)(-)M. q.e.d.

As a consequence of Theorem 9.7, we obtain a result of Matsu-shima [4].

Theorem 9.8. Let M be a Hodge manifold with a holomorphic vector field Z with empty zero set. Then M admits a holomorphic 1-form a such that a(Z)#0. In particular, the first Betti number b 1 of M is positive.

Proof By Theorem 9.7, Z is not in the ideal bi so that a(Z)#0 for some holomorphic 1-form a. q.e.d.

Theorem 9.4 is weaker than the result proved by Blanchard [1]. For the original theorem of Blanchard, see also Matsushima [5].

10. Zeros of Infinitesimal Isometries

Let M be an n-dimensional compact Kahler manifold and Z= X— iJX be a holomorphic vector field such that its real part X is an infinitesimal isometry. Assume that the zero set Zero(Z) (=Zero(X)) is nonempty. This is equivalent to assuming that g=clu, where c is the 1-form corresponding to X and u is a real valued function on M (see Corol-lary 4.6). We shall apply the Morse theory to the function u. A point of M is a critical point of u if and only if it is a zero point of X. Let Zero(X)= H Art be the decomposition of Zero(X) into its connected — components. Then each AT, is a closed totally geodesic submanifold of even co-dimension by Theorem 5.3 of Chapter II. Since X preserves the complex structure J, its covariant derivative VX commutes with J. If x eNt , then for a suitable orthonormal basis of Tx (M) the linear endo-

10. Zeros of Infinitesimal Isometries 113

morphisms (VX)„ and .1x of 7,',(M) are given by matrices of the form (see the proof of Theorem 5.3, Chapter II):

(VX)„ -

0

0 0 al

— al 0

\

0 ak

— 42k 0

,

! 0 1 —1 0

,

where ai #0 for i = 1, ... , k. The subspace of Tx (M) spanned by the first 2 n —2 k elements (resp. the last 2k elements) of the basis is the tangent space Tx (Ni) (resp. the normal space 71± (1Vi) to Ni). It follows that T(IV) is invariant by J so that Ni is a complex submanifold of M. Since du= J the Hessian matrix of u at x is given by J„ 0 (VX) x . The latter is a diagonal matrix with diagonal elements 0, ... , 0, — al , — al , ..., — ak , — ak . The nullity of the Hessian of u at x is equal to 2 n:— 2k and hence Ni is a non-degenerate critical submanifold. The index of the Hessian of u at x, i.e., the number of the negative diagonal entries in ./x 0 (VX)„, is clearly even since the nonzero diagonal elements appear in pairs.

We shall now sketch an outline of the proof of the following theorem of Frankel [1].

Theorem 10.1. Let X be an infinitesimal isometry on a compact Kahler manifold M with nonempty zero set Zero (X). Then for any coefficient field K,

E dim Hq (M; K). E dim Ilq (Zero (X); lq. q q

More precisely, if Zero (X). U Ni is the decomposition into connected components, then

dim 11q (M ; K)=Edim AI_ a ,(IVi ; K), i

where ili is the number of the negative eigen-values of J (V X)„ for x e and hence is an even integer.

Proof By Theorem 4.3, the complex vector field Z=X—iJX is holomorphic. To the function u constructed above, we apply the follow-ing result of Bott [3] :

Lemma. Let u be a real valued function on a manifold M such that M,= (xeM; u(x)r) is compact for every real number r. Let a < c<b be


real numbers such that c is the only critical value of u in the interval [a, b] and assume that N =u'(c) is a non-degenerate critical submanifold. If N =U N is the decomposition into connected components, then for any coefficient field K we have

Hq (Mb , M.; 1) .1dE Hq _ A (N ; K),

where A i is the index of the Hessian of u at a point of N.

The proof is similar to that of the case where N is a non-degenerate critical point (see Milnor [1]).

Let c 1 c2 ... be the critical values of u and ao , a1 , ... be real numbers such that ao < ci <a1 < c2 <a2 < • • •. Setting M p = Ma p = {xeM;u(x)a p } for p--=0, 1, 2, ..., we filter the group C(M) of singular chains by the sub-groups C(M). Then in the resulting spectral sequence, we have

Hp ,(Mp , ; K).

Applying Lemma 1 to the right hand side and writing q — p for r, we obtain

E= • IC). p,q- p q-Ai (N 1 ,

From dim H (M K)= dim E Eœ <dim E q P.(1 - 13 '

we obtain

dim Hq (M ; K)_Edim H q (1Vi ; K).

This establishes the inequality in one direction. The theorem follows from a result of Floyd [1] and Conner [1] (see Theorem 5.5, Chapter II), which gives the inequality in the other direction. q. e.d.

In Borel [3; p. 171], Theorem 10.1 is proved for a toral group acting on a homologically compact Kdhler manifold, e. g., a compact symplec-tic manifold (with a coefficient field K of characteristic 0).

The following corollaries are also due to Frankel.

Corollary 10.2. Let X be as in Theorem 10.1 with nonempty Zero(X). Then

(1) Zero(X) has torsion if and only if M has torsion.

(2) H21 , 1 (Zero (X); Z) =0 for all i if and only if H 2i +1 (M ; Z)= for all

Proof. (1) This follows from Theorem 10.1 and from the fact that a space has no torsion if and only if its rational Betti numbers coincide with its mod p Betti numbers for all primes p.

11. Zeros of Holomorphic Vector Fields 115

(2) Assume H2 i + 1(Z,ero(X); Z) = 0 for all i. Since each connected component Ni of Zero (X) is a complex submanifold and hence an orientable manifold of even dimension, the Poincaré duality theorem implies that Ni has no torsion. By (1) M has no torsion. Since the indices yli in Theorem 10.1 are all even, Theorem 10.1 implies that H2, + 1(M; Q) = 0 for all i. Hence, H2i+I (M; Z)=0 for all i. The implication in the other direction can be proved in a similar fashion. q. e. d.

Corollary 10.3. Let X be as in Theorem 10.1 with nonempty Zero(X). If Zero(X) is discrete, then M has no torsion and its odd dimensional Beta numbers vanish.

Recently, E.Wright [1] improved Corollary 103 by proving the following result.

Theorem 10.4. Let X be as in Theorem 10.1 with nonempty Zero(X). If Zero(X) is discrete, then

HP.q(M;C)=0 for p*q.

11. Zeros of Holomorphic Vector Fields

In this section, we shall describe recent results of Howard [1] on holo-morphic vector fields with nonempty zero set.

Theorem 11.1. Let M be a compact Kahler manifold with a nontrivial holomorphic vector field Z such that Zero (Z) is nonempty. Then

1-1"'° (M ; C) = 0 (n= dim M),

that is, M admits no nonzero holomorphic n-forms.

Proof Let C be the (0, 1)-form corresponding to Z. Since Zero(Z) is nonempty, there is a function f on M such that C =d"f (see Corollary 4.5). Since

Z(f) = d 'AZ) = C (4 = 11 Z 11 2 9

Z (f) 0 and >0 almost everywhere. Let 9 be a holomorphic n-form on M. Since every holomorphic

form on M is necessarily closed, we obtain

Lz 9=do t z 9+t z od9=0.

Since 4=(d9)=0 and tz =0, we obtain

Lz rp=dot z rp-Ft z odrp=0.

Hence, L z ((p A (7).o.


Making use of this formula, we obtain

Z(f)•4 yN=L z (f7. 9 A -f L z (9 i.\)=Lz (f. A4

Since f. 9 A 0 is a form of degree (n, n), it is closed and hence

Lz (f9 A ço)=d o t z (f. ( A -0)=

by Stokes' theorem. Hence,

Z( (in2 9 A 0=0.

(The factor in2 makes the (n, n)-form in2 Q y.‘i) real and non-negative.) Since Z(f) ,--- I1Z112 is positive almost everywhere, we conclude that 9 vanishes identically on M. q.e.d.

Since a holomorphic n-form is a holomorphic section of the canoni-cal line bundle K, the following theorem generalizes that of Howard.

Theorem 11.2. Let M be a compact Kiihler manifold with a nontrivial holomorphic vector field Z such that Zero (Z) is nonempty. Let K be the canonical line bundle of M and p be a positive integer. Then the line bundle KP admits no nonzero holomorphic sections.

Proof Let = d"f be the (0, 1)-form corresponding to Z as in the proof of Theorem 11.1 so that Z(/)= II Z11 2 . Let 9 be a holomorphic section of KP. As in the proof of Theorem 2.1, in terms of a local coordi-nate system 9 may be symbolically expressed as follows:

= h(di A • • • A d?)", where h is a holomorphic function defined in the coordinate neighbor-hood. Define a real, non-negative (n, n)-form 19 I 2/P by

I91 2/P = r2 In the proof of Theorem 2.1 we established that the largest connected group of holomorphic transformations of M leaves every holomorphic section of IQ invariant Hence 1912/P is invariant by the 1-parameter group generated by Z, that is, Lz (191 2/P). O. The remainder of the proof is essentially the same as the proof of Theorem 11.1. Since

Z019113/2 = Lz(1149r 2)-1 Lz(k'r 2)=Lz(f 191P12)

=do tz(f 1(pI P12).

Integrating the both ends of the equalities above and making use of Stokes' theorem, we obtain

Z(/)191 142 =O.

dz i A •-• A de A cre A • • • A dr

11. Zeros of Holomorphic Vector Fields 117

Since Z(t). Zil 2 is positive almost everywhere, we conclude that (I pIP I 2 and hence ço vanish identically on M. q.e.d.

The following theorem of Howard sharpens Theorem 11.1 for Hodge manifolds.

Theorem 11.3. Let M be a Hodge manifold with a nontrivial holo-morphic vector field Z such that Zero (Z) is nonempty. Then M admits no nonzero holomorphic p-form for p> dim Zero (Z).

The set Zero(Z) is a subvariety (possibly with singularities) of M, and by dim Zero (Z) we mean the maximum dimension of its compo-nents.

Proof Let n = dim M. By Theorem 9.4, there is an imbedding of M into a complex projective space PN (C) such that the holomorphic vector field Z can be extended to a holomorphic vector field, denoted also by Z, of PN (C). Since the automorphism group of PN (C) is finitely covered by the group SL(N +1; C), Z may be considered as an element of the Lie algebra of SL (N +1; C). With respect to a suitable basis of CN + 1, Z is represented by a triangular matrix, and let z°, ziv be the coordi-nate system in CN + 1 with respect to such a basis. Let Lk be the k-dimen-sional linear subspace of PN (C) defined by

Then Z is tangent to Lk at each point of Lk for k =0, 1, , N. Let a) denote the Kahler (1, 1)-form of PN (C) corresponding to an invariant Kahler metric. Multiplying w by a suitable positive constant, we may assume that w represents the characteristic class of the line bundle over PN (C) defined by a hyperplane. Thus w represents the second cohomology class dual to the 2(n-1)-dimensional homology class represented by a hyperplane.

Lemma 1. Let M and Z be as above. If p is a nonzero holomorphic p-form on M, then there exist r-dimensional irreducible subvarieties If, of M for r =n, n— 1, , p with the following properties:

(1) Z is tangent to n at every regular point of V,. (2) If j,: 17,—)M denotes the imbedding and [cp A P] denotes the

cohomology class in H2 ' P (M ; C) represented by the (r,r —p)-form rp A P, then 4 [q• A_P] is a nonzero element of 112 r - P(K; C).

The proof of Lemma 1 is by induction on r starting with r=n. For r= n, it suffices to take K= M. It is well known that the multiplication by (On- P defines an isomorphism of HP(M; C) onto 1/2 ""(M; C) (see for example Weil [1; p.75]). Hence Up A CO" - II] is nonzero. Assuming Lemma 1 for r, let k be the smallest integer such that Lk contains 1/,..


Viewing V, as an algebraic variety in the k-dimensional projective space Lk , we consider the hyperplane section V, n Lk _ i . Since co is the charac-teristic class of the line bundle defined by a hyperplane, we obtain

[9 A al- P] I(Vr 4-1)=[(1) A al- 11 I Vr•

The right hand side is nonzero by assumption. We write V, n Lk_i = E m i Di , where Di's are irreducible divisors of V, and mi's are integers. Since [9 A for - 1- P] I (V, n Lk _ i)*0, it follows that [(I) A of -1- I Di * 0 for some Di . We denote such a divisor Di by I'. clearly J'

the properties (1) and (2) for r — 1. This completes the proof of Lemma 1.

Let V be the p-dimensional irreducible subvariety vp obtained in Lemma 1 and V' be the set of regular points of V. Denote the inclusion map V'—>M by j. Then as in the proof of Theorem 11.1, we have

Z(f) • j*(9A0)=j* L z (f7pArp)=j*dot z (f(pnço)+j* tz o d(Ao Ac -p-)

j* do t z (fp ço).

Now we make use of the following Stokes theorem (see P. Lelong [1; § 6]):

Lemma 2. Let V be a p-dimensional st4bvariety of a compact complex manifold M and V' be the set of regular points of V. Let 0 be a (2p —1)- form defined in a neighborhood of V. Then

d0=0.

Hence, V .

Z(f) • j* ((p A F.p)= j* d 0 t z (ho rp)= O.

As in the proof of Theorem 11.1, we conclude that 9=0 since Z cannot vanish identically on V' by our assumption dim Zero (Z)< p =dim V'.

Corollary 11.4. Let M be a Hodge manifold with a holomorphic vector field Z whose zero set Zero(Z) is nonempty and discrete. Then

(1) M admits no nonzero holomorphic p-forms for p >0; (2) the arithmetic genus of M is equal to 1;

(3) the fundamental group ni (M) has no proper subgroup of finite index and Hl (M; Z) ---- 0.

Proof (1) is immediate from Theorem 11.3. (2) follows also imme-diately since the arithmetic genus of M is equal to

E( —1)P dim HP. ° (M; C)= dim H°. ° (M; C)= 1 .

To prove (3), assume that ni (M) has a subgroup of finite index k and let it71 be the corresponding k-sheeted covering space of M. Lift Z to a

12. Holomorphic Vector Fields and Characteristic Numbers 119

holomorphic vector field 2 of ia and apply (2) to A. Then the arithmetic genus of /a is also 1. On the other hand, the Riemann-Roch-Hirzebruch theorem implies that the arithmetic genus of /a is k times that of M. Hence, k =1. Since Hi (M; Z)=ni (M)1[7r 1 (M), ni (M)] and Hi (M; R). 0, it follows that [ni (M), 7E 1 (M)] is a subgroup of ni (M) of finite index and hence, Hi (M; Z)= O. q. e. d.

(3) of the corollary above is due to Lichnerowicz [8]. He has shown many other results on zeros of holomorphic vector fields on compact Kahler manifolds with non-negative first Chem class.

12. Holomorphic Vector Fields and Characteristic Numbers

Let M be an n-dimensional compact complex manifold and Z be a holomorphic vector field on M. Let Zero (Z)=U 1Vi be the decomposi-tion of the zero set of Z into its connected components Ni . We assume that Z is non-degenerate along iv, in the following sense. Since we con-sider one iv, at a time, we denote iv, by N. First, we assume that N is a complex submanifold of codimension r so that dim N=n—r. Let Ti .°(M) and T1 ' ° (N) be the holomorphic vector bundles of complex vectors of type (1, 0) over M and N, respectively. We set

E . mi.° o (N) . I (MIN, E` =

Then E is a holomorphic vector bundle of rank n over N and E' is a holomorphic vector subbundle of E of rank n — r. The holomorphic vector field Z induces an endomorphism of the bundle E; in terms of a local coordinate system zl, , zn of M it is given by the matrix

a (acc y a zi) if Z= E —. We denote this holomorphic endomorphism 0 zOE

of E by A. (Since Z vanishes on N, the matrix (acc zP) defines A inde-pendently of the coordinate system.) The kernel of A contains E'. We assume that E' is exactly the kernel of A. Then denoting the image AEcE by E", we obtain a decomposition

E= E' ()E".

Thus, E is holomorphically isomorphic to the direct sum of two holo-morphic vector subbundles E' and E". We denote the restriction of A to E" by A. Then A is an automorphism of E". Let h' and h" be hermi-tian fibre metrics in E' and E", respectively. We extend the hermitian fibre metric h' +h" of E to a hermitian metric h of M.

We follow now closely § 6 of Chapter II. Let P be the bundle of unitary frames over M; it is a principal bundle over M with group U(n).


Let PM be the bundle of adapted frames over N; it is a principal bundle over N with group U (n — r) .x U (r). (By an adapted frame we mean a unitary frame whose first n—r basis elements are in E' and whose last r elements are in E".) Considering the hermitian connection of M, let Q be the curvature form on P. Then its restriction to 4, is of the form

IQ, 0\ ko Qv !'

where, with an obvious identification, Q the curvature form for h' and Qv is the curvature form for h".

Let V denote the covariant differentiation of the hermitian connec-tion defined by h. As in § 4, we write

VZ=V/Z+V"Z,

where V' Z and V"Z are defined by the property that

Viv Z=0 and V,1;, Z=0 for all vectors Wof type (1, 0).

Then the endomorphism A of E coincides with the restriction of V' Z to N. As in §6 of Chapter II, we may consider V1 Z as a tensorial 0-form of type ad( U(n)). Restricted to 4,, v'Z is of the form

/0 0\ \0 A

Let f be an ad(U(n))-invariant symmetric form of degree n on the Lie algebra u(n). The simplest example is given by det (=determinant). As in §6 of Chapter II, we define the residue Resf (N) by

f(ti2+SPZ) Resf (N) • tn - ' =

det(tS2,+A)'

where the bars over f and det indicate that the forms are pull down to N. Now the theorem of Bott [1, 2] may be stated as follows:

Theorem 12.1. Let M be a compact complex manifold of dimension n. Let Z be a holomorphic vector field of M with zero set Zero(Z)= H N - where the Ni's are the connected components of Zero(Z). Let f be an (ad U(n))-invariant symmetric form of degree n on u(n). Then the charac-teristic number I f(Q) of M defined by f is given by

SAM= E Resf (Ni), i

provided that Z is non-degenerate along Zero (Z) in the sense defined above.

12. Holomorphic Vector Fields and Characteristic Numbers 121

The proof is almost identical to that of Theorem 6.1 of Chapter II and hence is omitted. We remark that A= VT ZI N is holomorphic and hence the coefficient of tk in f(tf2+V"Z)I N is a polynomial in the curva-ture form with constant coefficients since a holomorphic function on a compact manifold must be constant. (In § 6 of Chapter II, a similar fact was derived from the property that VX is parallel on N.)

In Bott [1], the theorem is proved when Zero(Z) consists of isolated points. The general case is proved in Bott [2]. A generalization to meromorphic vector fields has been obtained by Baum-Bott [1]. For a different proof, see Atiyah-Singer [1]. When Zero(Z) consists of isolated points, the theorem follows also from a Lefschetz formula of Atiyah-Bott [1]. See also Illusie [1].

Corollary 12.2. If a compact complex manifold admits a holomorphic vector field with empty zero set, then its Chern numbers vanish.

IV. Affine, Conformal and Projective Transformations

1. The Group of Affine Transformations of an Affinely Connected Manifold

Let M be a manifold with an affine connection and L(M) be the bundle of linear frames over M. Let 0 and co denote the canonical form and the connection form on L(M), respectively. We recall (§ 1 of Chapter II) that a transformation f of M is said to be affine if the induced automorphism f of L(M) leaves co invariant. We quote the following result established earlier (see Theorem 1.3 of Chapter II).

Theorem 1.1. Let M be an n-dimensional manifold with an affine connection. Then the group 91 (M) of affine transformations of M is a Lie transformation group of dimension ..n(n+ 1).

As in § 3 of Chapter II, it is natural to ask when the maximum dimen-sion n(n + 1) is attained. We prove

Theorem 1.2. Let M and 91(M) be as in Theorem 1.1. Then dim 91 (M) = n (n +1) if and only if M is an ordinary affine space with the natural flat affine connection.

Proof Assume dim 9.1[(M)= n(n+ 1). From Theorem 1.3 of Chapter II, it is clear that the identity component 9I°(M) acts simply transitively on each connected component of L(M). This implies that every standard horizontal vector field 5C (1. e., vector field iC such that (Da) =0 and 000=a constant element in R") on L(M) is complete; the proof is similar to that of Theorem 2.5 of Chapter II. This means that the connec-tion is complete (see Proposition 6.5 of Chapter III in Kobayashi-Nomizu [1]).

Let 91„ be the isotropy subgroup of 91(M) at a point xeM. Since dim 9I„ = n 2 , 9I„ contains GL + (n; R), where

(n; R) = {aeGL(n; R); det a>0}.

(Identifying 9Ix with the linear isotropy group, we consider it as a subgroup of GL(n; R) by choosing a basis in the tangent space Tx (M).) In

1. The Group of Affine Transformations 123

particular, it contains homothetic transformations t In with t > O. If K is a tensor of contravariant degree p and covariant degree q at the point x, the transformation t in sends K into K. Since the curvature tensor field R is of contravariant degree 1 and covariant degree 3, the transfor-mation tin sends Rx into t - 2 Rx . On the other hand, 91„ must leave Rx invariant so that Rx = r 2 Rx for all t> O. Hence, Rx = O. This shows that the curvature tensor field vanishes identically. Similarly, the torsion tensor field vanishes identically. Hence, the connection is flat.

Let A4 be the universal covering space of M. Since AI is a simply connected manifold with a complete flat affine connection, it is an ordinary affine space with the natural flat affine connection (see Theo-rem 7.8 of Chapter VI in Kobayashi-Nomizu [1]). From the fact that no element of 91(k) other than the identity commutes with 91° (k) element-wise, we can conclude that M itself is simply connected as in the proof of Theorem 3.1 of Chapter II.

Finally, the converse is evident. q.e.d.

The local version of Theorem 1.2 is classical; see, for example, Eisen-hart [1].

The following theorem is due to Egorov [1] (see also Yano [3]).

Theorem 1.3. Let M be an n-dimensional manifold with an affine connec-tion and 91(M) be the group of affine transformations. if dim 91(M) > n 2, then the connection has no torsion.

Proof Assuming that the torsion tensor does not vanish at a point x, we shall show that the dimension of the isotropy subgroup 21„ is at most n2 — n. Since 9.1„ leaves the torsion tensor at x invariant, it suffices to prove the following algebraic lemma.

Lemma. Let V be an n-dimensional vector space and T be a nonzero element of V () A 2 V*, where V* is the dual space of V. Let G be the group of linear transformations of V leaving T invariant. Then

dim G< n 2 n.

Proof of Lemma. We consider Tas a skew-symmetric bilinear mapping V x V—) V Let X: V—) V be a linear transformation and 91 = et x be the 1-parameter group of linear transformations generated by X. Then X is in the Lie algebra g of G if and only if

T(9, v, 9, w)= 9,(T(v, w)) for v, we V and all teR.

Differentiating this equation with respect to t at t= 0, we see that X is in g if and only if

T(X y, w)+ T(v , X w)= W (v , w)) for y, we V.

124 IV. Affine, Conformal and Projective Transformations

Take a basis in 17 and let Tiik and Xi be the components of T and X with respect to the chosen basis. Then the equation above is equivalent to the following system of linear equations:

E (TIc xi+ Tbai — xi Tbjc)=o for a,b, c=1, n.

This can be rewritten as follows:

0 (a,b,c=1,...,n), where

aj i= Ta c Ta b i 61 — Tic (51.

This is a system of linear equations with n 2 unknowns Xj and coefficients i . If we can find n linearly independent equations in this system, then

we will know that the dimension of the space of solutions of this system does not exceed n 2 n. Let T:c be one of the non-vanishing components of T Since Tik is skew-symmetric in the lower indices, Tba,*0 implies b * c. By reordering the basis if necessary, we may assume that either a = 1, b = 2, c= 3, or a=b=-1, c=2, i.e., Th 0 or T112 +0.

We shall first consider the case where T21 3 + 0. We claim that the n linear equations given by

E Xj= 0 k=1, ...,n

are linearly independent. In fact, consider the n x n matrix (An defined by 4= Ak213 . Then

TA 61+ ni 6; — — 713 (51:.

Since Th + 0, the matrix (4) is non-singular. Next, assume T112 +0. We claim that the n linear equations given by

EA1LX ii ==0

EAui xii =o k=2,..., n,

are linearly independent. In fact, consider the n x n matrix (Ai) defined by

= A-12 2 = T 2 22 + T122 6i2 2 6i= T 2 12 (5i I — 2

Alc= A112 = Vic (5.-i — Ti k 6 1 •31 + pi J

for j= 1, , n and k= 2, ... , n.

Then this matrix is of the following form:

— * 0 Tj2

2. Affine Transformations of Riemannian Manifolds 125

where in _ 1 denotes the identity matrix of order n — 1. This matrix is clearly non-singular. q. e.d.

We state Another result of Egorov [3] which can be proved by a similar method (see also Yano [3]).

Theorem 1.4. Let M and 91(M) be as in Theorem 1.3. If dim 91(M) > n2, then the connection has neither torsion nor curvature, j. e., it is flat, provided n.. 4.

Remark. In connection with Theorem 1.3, Egorov [5] (see also Yano [3]) has shown that if dim 91(M)= n 2 and n 4, then there exists a 1-form T on M such that the torsion tensor T is given by

T(X, Y)= t (Y) X — T (X) Y for all vector fields X, Y.

For example, define an affine connection in M =Rn in terms of the natural coordinate system x', ..., xn and the Christoffel symbols as follows: EA= —61, r=o for j#1,

so that its torsion T can be constructed from the 1-form t = d x l in the manner described above. Since the group 91(M) contains the translations in Rn, it is transitive on M =Rn and its isotropy subgroup 910 at the origin is the group of linear transformations of Rn leaving the point x' =1, x 2 = • • • ---- xn = 0 invariant. Hence, dim 91(M)= n 2 . Clearly, the curvature of this connection vanishes identically.

If we symmetrize the affine connection above by setting

rtkk =r;,ii = --pi, r=o for j, k # 1,

then we obtain a torsionfree affine connection with nontrivial curvature eln M = W. The group 91(M) of affine transformations for this connection is the same group of dimension n 2 .

Wang and Yano [1] have determined the n-dimensional affinely connected manifolds such that dim 91(M) > n 2 — n +5. For more details on related results, see Yano [3].

2. Affine Transformations of Riemannian Manifolds

In this section we shall compare the group 3(M) of isometries and the group 91(M) of affine transformations of a Riemannian manifold M.

The following result is due to Hanno [1]. See also Kobayashi-Nomizu [1, vol. 1; p. 240].

Theorem 2.1. Let M = Mo x Mi x • • • x Mk be the de Rham decomposi-tion of a complete, simply connected Riemannian manifold M, where M 0 is


Euclidean and M1 , ..., Mk are all irreducible. Then

91° (M ) = 9-01 4 o) x 91° (A 4 1) x ... x

3° (M)= 3 ° (M 0) X 3Cb (M1 ) X — X

Since the group of affine transformations and that of isometrics of a Eulcidean space are well understood, the study of 21°(M) and 3°(M) is essentially reduced to the case where M is irreducible. For a proof of the following result, the reader is again referred to Kobayashi-Nomizu [1, vol. 1; p. 242].

Theorem 2.2. If M is a complete, irreducible Riemannian manifold, then 9.1(M)= 3(M) except when M is a 1-dimensional Euclidean space.

Let M be a complete Riemannian manifold and AI be its universal covering space. Let ft-4 = Mo x Mi x • • • X Mk be the de Rham decomposi-tion of A 1 . By Theorem 2.1, the Lie algebra a(S1) of 91(k) is isomorphic to a (Mo)+ a (MI ) + • • • + a (Mk ). Let X be an infinitesimal affine transfor-mation of M and it be its natural lift to M". Let (X0 , X1 , ... , Xk) be the element of a (M0) + a (M1 ) + • • • + a (Mk) corresponding to .)-C E a (M). By Theorem 2.2, X1 , ..., Xk are all infinitesimal isometrics. If Al has no Euclidean factor M0 , then X0 is zero so that it is an infinitesimal isometry. If 'it. is an infinitesimal isometry, so is X. The assumption that Mo be trivial can be expressed in terms of the restricted linear holonomy group. We can therefore state the result as follows (see also Lichnerowicz [1; p. 83], Yano-Nagano [3]).

Corollary 2.3. If M is a complete Riemannian manifold such that its restricted linear holonomy group W °(x) leaves no nonzero vector at x fixed, then 2e(M)=3 °(M).

The assumption in Corollary 2.3 is technically a little stronger than the condition that M admits no nonzero parallel vector field. The latter condition amounts to assuming that the linear holonomy group tri(x) leaves no nonzero vector at x fixed.

With the notation above, if the length of an infinitesimal affine trans-formation X of M is bounded, the same is true for X0 , X1 , ... , X. But an infinitesimal affine transformation X0 of the Euclidean space Mo is of bounded length if and only if X0 is an infinitesimal translation which is a very special kind of infinitesimal isometry (see Kobayashi-Nomizu [1, vol. 1; p. 244]). Hence (Hano [1]),

Corollary 2.4. If X is an infinitesimal affine transformation of a com-plete Riemannian manifold with bounded length, then it is an infinitesimal isometry.

3. Cartan Connecticins 127

In particular, we rediscover Corollary 2.4 of Chapter 11 due to Yano.

Corollary 23. If M is a compact Riemannian manifold, then 91°(M)-- 3°(M).

3. Cartan Connections

In this section we shall treat conformal and projective connections in a unified manner.

Let M be a manifold of dimension n, .2 a Lie group, 20 a closed sub-group of 2 with dim 2/20 =n and P a principal bundle over M with group 20 . We give a few examples:

Example 3.1. Let .2 be a Lie group and 20 a closed subgroup of Q. Set M = 2/20 and P== Q.

Example 3.2. Let 2 be the affine group 9I(n) acting on an n-dimen-sional affine space and 20 = GL(n; R) an isotropy subgroup of 2 so that 2/20 is the affine space. Let M be a manifold of dimension n and P the bundle of linear frames over M.

Example 3.3. Let .2 be the group of Euclidean motions of an n-dimen-sional Euclidean space and 20 =0(n) an isotropy subgroup of 2 so that 2/20 is the Euclidean space. Let M be a Riemannian manifold of dimen-sion n and P the bundle of orthonormal frames over M.

Example 3.4. Let 2 be the projective general linear group PGL(n; R) acting on an n-dimensional real projective space and 20 an isotropy sub-group of 2 so that 2/20 is the projective space. Let M be an n-dimen-sional manifold. We shall later construct a principal bundle P over M with group 20, called a projective structure.

Example 33. Let 2 be the Möbius group 0(n +1, 1) acting on an n-dimensional sphere S" and 20 an isotropy subgroup of .2 so that 2/20 is the sphere S. (This will be explained in detail later.) Let M be an n-dimensional manifold with a conformal structure and P be the first prolongation of the conformal structure. (This will be also explained in detail later.)

Since 20 acts on P on the right, every element A of the Lie algebra 10 of 20 defines a vertical vector field on P, called the fundamental vector field corresponding to A (Kobayashi-Nomizu [1, vol. 1; p. 51]). This vector field will be denoted by A*. For each element a of 20 , the right translation by a acting on P will be denoted by Ra . A Cartan connection in the bundle P is a 1-form w on P with values in the Lie algebra I of 2 satisfying the following conditions:


(a) co(A*)= A for every A E 10 ;

(b) (R.)* = ad (a -1 ) co for every element a e 20 , where ad (a") is the adjoint action of a' on I;

(c) w(X) * 0 for every nonzero vector X of P. Condition (c) means that a) defines a linear isomorphism of the tangent

space (P) onto the Lie algebra I for every ueP since dim P. dim Q. In other words, co defines an absolute parallelism on P.

A Cartan connection in P is not a connection in P in the usual sense, for a) is not I0-valued. It can be, however, considered as a connection in a larger bundle /32 obtained by enlarging the structure group of P to 2, i.e.,

P2 =Px 20 .2.

Then P is a subbundle of P2 and a Cartan connection a) in P can be uniquely extended to a usual connection form on P2, also denoted by a). (Take, for instance, Example 3.2. Then an affine connection of M is a Cartan connection in P. On the other hand, a linear connection of M is an ordinary connection in P. For more details on this point, see Koba-yashi-Nomizu [1; Chapter III, §§ 2-3].)

If we set E=P2/.20 , then E is the bundle with fibre 2/20 associated with P. We can identify M with the image of the natural mapping P P2/20 . In other words, we have a natural cross section M — E. In Examples 3.2 and 3.3, E is the tangent bundle of M and the natural cross section is the zero section. In Example 3.1, E is the product bundle M x M and the natural cross section M E. M x M is the diagonal map. Geo-metrically speaking, condition (c) means that the fibre of E over each point xeM is tangent to M at x, see Ehresmann [1], Kobayashi [4]. But this geometric interpretation of (c) will not be used here.

We define the curvature form 0 of the Cartan connection a) by the following structure equation:

dco. — [co, co] +O.

It is an I-valued 2-form on P. For instance, if a) is an affine connection, then Q is a 2-form with values in the Lie algebra a(n) of the group 91(n) of affine motions. Decompose a (n) into the vector space direct sum of the translation part Rn and the linear transformation part gl(n; R). Then the Rn-component of Q is the usual torsion form and the gl (n; R)-component of Q is the usual curvature form of the corresponding linear connection, see Kobayashi-Nomizu [1; § 3 of Chapter III].

Given a Cartan connection co in P, we call a transformation q) of P an automorphism of (P, co) if it is a bundle automorphism, j. e., commutes with the right translations Ra , (ae 20), and if it preserves the form a).

3. Cartan Connections 129

From Theorem 3.2 of Chapter I we obtain the following result (Koba-yashi [4]):

Theorem 3.1. Let co be a Cartan connection in P. Then the group 91(P, co) of automorphisms of (P, co) is a Lie group with dim 9I(P, a)). dim P.

For a fixed point u0 of P, the mapping Qe 91 (P, co)—> cp(u 0)eP imbeds 91(P, w) as a closed submanifold of P.

From now on we shall assume that the Lie algebra I of .2 is graded as follows:

1 ==g-i+go+gi+•••+9k (a vector space direct sum), with

Egi, g] c91 and lo = go + + • • • + gk •

Let w=a)--i+wo+col+•••+cok

be the corresponding decomposition of the Cartan connection w. Since w defines an absolute parallelism on P, the algebra of differential forms on P is generated by w (i. e., the components of w with respect to a basis for I) and functions on P. But the curvature Q of w does not involve co o , W 1 , , wk . In fact, this is a direct consequence of the following three facts:

(i) The g_ 1 -component w_ 1 , restricted to each fibre of P, vanishes identically.

(ii) The I0-component w 0 + w1 + • • • + Wk, restricted to each fibre of P, is the Maurer-Cartan form and hence defines an absolute parallelism.

(iii) The curvature Q, restricted to each fibre of P, vanishes identically. Condition (a) for Cartan connections implies both (i) and (ii). To

prove (iii), it suffices to observe that the structure equation of the connec-tion restricted to a fibre gives the structure equation of Maurer-Cartan for the group 20 and then apply (i).

If we choose a basis el , , en for g_ 1 and write

w_ i =wi el +••• +wn en ,

then we can express the curvature Q of the Cartan connection w as follows:

Q=ElKii co i nari,

where each Ku is an I-valued function on P.

Theorem 3.2. Let .2 be a Lie group and 20 a closed subgroup of .2 such that the Lie algebra I o of .20 contains an element E with the property that the Lie algebra I of .2 is graded as follows:

I= 9 -1 + go + + • • • + 9k

130

IV. Affine, Conformal and Projective Transformations

where gr= {X el; [E, X] = r X} for r= —1, 0, 1, , k

Let P be a principal 20-bundle over M and co a Cartan connection in P (with values in 1). If, the automorphism group 91(P, co) has the maximum dimension, i. e., dim 91(P, co)= dim P, then the curvature Q of co vanishes identically.

Proof: Set a, = exp(tE)e 2 0 . Fix a point 140 e P and let ço, be an element of 91(P, co) such that Rat u0 =q),(u 0). Since the orbit of 9.1(P, co) through 140 is closed and has the same dimension as P, such an element (p, exists. Let Q, be the g,-component of S2 and write

fl„ =E4 'Cry w t

Since ço, leaves w and S2r invariant, it leaves K ru invariant. We shall compare this action of ço, with that of R. t . Since ad (E) coincides with the multiplication by r on g„ ad(a,) is the multiplication by en on g,.. Hence,

Ra*,(S r) = ad (a7 1 ) r = " Q„

Rt(co_ 1 )=ad(a,-1 )co_ 1 =e1 (0_ 1 .

From these two equalities, we obtain

e- ri K rii coi=e- rt Or = Rt(Or)

=E Rt(K ri) ncoi A R:coi

=E4 R(K ru) e2 f cd A a). Hence,

Rt (K ru)=6,-(r+ 2)1 Kru . On the other hand,

(e (KO = K r ij •

Comparing these two equalities at 140 , we obtain

u (u0) = K, u(q), u 0)= K r ii (Rat u0) = e (r+ 2 " K ru (u0) .

Hence, Krii (u0)= O. Since uo is an arbitrary point of P, this proves our assertion. q.e.d.

If we apply Theorem 3.2 to Examples 3.2 and 3.3, we obtain local versions of Theorem 3.1 of Chapter II and Theorem 1.2. (Note that l= g_ 1 +g, in Examples 3.2 and 3.3 and the.g_ 1 -component of Q is the torsion form while the go-component of S2 is the curvature form in the usual sense.)

For this section, see also Ogiue [2].

4. Projective and Conformal Connections 131

4. Projective and Conformal Connections

In the preceding section, we studied Cartan connections in general. In this section we shall describe both projective and conformal connections in a unified manner. We begin with a simple algebraic proposition (see Kobayashi-Nagano [1]).

Proposition 4.1. Let 1= g_ 1 + go + g1 + • • • +gk be a semisimple graded Lie algebra (with g _ 1 * 0 and go #0). Then

(1) I=9-1+g0+ 91, i.e., g 2 =O. (2) The linear end omorphism a of1 defined by

a(X_ i + X0 + Xl. )= X_ i + XIL for Xi E gi

is an involutive automorphism of 1. (3) With respect to the Killing-Cartan form B of!, (i) g_ 1 + g1 is perpendicular to g0 ; (ii) Blg_ i =0 and Blgi =0; (iii) g1 is the dual vector space ofg_ 1 under the dual pairing (X_ 1 ,

B(X_ i , X1 ). (4) The two representations ad1 (g0)1 g_ and adi (go) I gi of g0 are dual

to each other with respect to the Killing-Cartan form B. (5) There is a unique element Eego such that

gi ={Xel; [E, X]=iX) i= —1,0,1.

Proof (1) Let Xegi and Z eg2 . Since (ad X)(ad Z) maps gi into gi+i , 2 and i+ 2 1 so that j*j+ i+ 2, the trace of (ad X) (ad Z) is zero. Hence, B(X , Z)= 0. Since B is non-degenerate, it follows that Z=0.

(2) The proof is straightforward. (3) Since B is invariant by the automorphism a, we have

B(X_ 1 + X1 , X0) =B (a (X_ 1 + X1 ), a (X0)) = —B(X_ 1 + X1 , X0),

which proves (i). To prove (ii), let X_ 1 , Y_ 1 eg_ i . Then (ad X_ 1 ) (ad Y_ 1 ) maps gi into g._ 2 and hence its trace is zero. This shows that

Y_ 1 )=0.

Similarly, B I gi =0. To prove (iii), let X_ 1 eg_ i and assume B(X_ , g1 )-0. By (i) and (ii), this implies B(X_ 1 ,I)=0. Hence, X_ 1 =0. Similarly, if

e gi and B(X it , g_ 1 )=0, then X1 =0. (4) Let X0 ego . Since B is invariant by ad X0 , we have

B (ad (X0) X_ 1 , X 1 )= — B(X_ 1 , ad (X0) X1 ) for Xi egi , i= +1.


(5) Let E be the linear endomorphism of I defined by

a(Xi)=iXi for Xi egt , i= —1,0, 1.

Then s is a derivation of the Lie algebra. Since every derivation of a semi-simple Lie algebra is inner, there is a unique element Eel such that ad(E)= i. To prove that E is in go , we observe that ad(ot(E)) coincides with ad (E). Hence, Œ (E)=: E. By the definition of a, E ego . q.e. d.

The graded Lie algebras of Proposition 4.1 have been classified by Kobayashi-Nagano [1]. We are interested here only in the following two example. For a general theory, see Ochiai [3].

Example 4.1. Let 2/20 be the real projective space of dimension n, where

2= PGL(n; R)= SL(n +1; R)/center,

{ A 0 Qo = (

a) ESL(n + 1 ; R) } /center,

where A eGL(n; R) and is a row n-vector,

y 21 = { (

I 1 0

) ; c: row n-vector } .

The graded Lie algebra I= g_ ii + go +g1 with this .2/20 is given by

1=5I(n+1; R),

9 - 1 = { Co ov) } , A 0

go= I( 0 a); trace A + a=0}, g1=

{(0 00» ,

where y is a column n-vector, is a row n-vector, A egl(n; R) and aeR. The element Eeg o is given by

lain 0\ E= k o bi '

We may describe the graded Lie algebra I as follows. Let 1/ be the n-dimen-sional vector space of column n-vectors and 1/* be the dual space con-sisting of row n-vectors. Then

I= 17 -1-gI(n; R)+ 1/* under the identification

/0 y) (0 0) eg i _—>ye1/*, egi —> E V*,

k0 0 0

/A 0\ ego —>A—ain egl(n;R). k o al

where a= — 1/n +1, b = nIn +1 .


Under this identification, the Lie algebra structure in I is given by

[y, V] =0, [, ']=0; [U, v]= U v, [, U]= U,

[U,UT.-=UU'—Ui U,

where y, We V, , 'E V*, U, U'egl(n; R). Clearly, the element E is now given by — in egI(n; R). With respect to the natural bases el , ... , en of V, e1 , ... , e of V* and ei of gl(n; R), we write the Maurer-Cartan form co of PGL(n; R) as follows:

co =Eal ei +Ea4 el +Ecoi ei .

Then the Maurer-Cartan structure equations of PGL(n; R) are given by

d al = -E Cdk A cok ,

d co. I = —E al k A co., — coi A a) i + (5.ii Ecok A cok ,

dcoi = —Ecok A ail.

Example 4.2. We describe the n-dimensional Möbius space Sn, first geometrically and then group-theoretically. Let S be the symmetric matrix of order n +2 given by

S=

(

0 0 — 1 0 in 0).

\—i 0 0

Let xeRn+ 2 be a nonzero column vector, considered as a point in the real projective space g+i (R). The quadric Sn in g + ,(R) defined by

ixSx=0

is the n-dimensional Möbius space. It is diffeomorphic to the n-sphere (y1)2 ± ... _F un2 +1• , ) 1 in W +1 under the mapping defined in terms of the natural homogeneous coordinate system x°, x1 , ..., xn + 1 of 'F1 (R) by

x°=. 1 (1— yn+i), xn± 1 .--1(1 + yn + 1 ).

Let p be the natural projection from Rn + 2 — {0) onto R +1 (R) and ds 2 be the natural Riemannian metric on g + , (R) defined by

p* (d s 2) = 2 [(E xi xi) (E d x i d A—( x dx 1) 2 } /(E x i xi) 2 .

Then the diffeomorphism above is an isometric mapping of the unit sphere onto the Möbius space sr!. Let

2=0(n+1,1)={XeGL(n+2; R); 1XSX=S},

.20 = (a-1 0 0

{ v A +0(n+1, b a

1); Ae0(n) aeR, eR''} ,


where c is written as a row vector. By a simple calculation it can be shown that

v=a -1 A•`, b=(2a) -1 •. Let

21.=.1(1. j,, 0O0

)e0(n+1, 1); 'eltn}, b 1

Then 2 acts transitively on the Möbius space Sn with 20 as the isotropy subgroup at the point defined by x° = x1 ---- • • • = xn = 0, called the origin of the Möbius space S". The subgroup 21 is the kernel of the linear isotropy representation of 20 at the origin. The graded Lie algebra I= g_ 1 + go +git associated with 2/20 is given by

I=o(n+1,1)=Vegl(n+2; R); 'XS +SX=0),

0 tv 0 9_ 1 = {(0 0 V)

0 0 0

(—a 0 0

go--- { - 0 A 0); Aeo(n)}, 0 0 a

0 0 0 gl = {(1. 0 0 )1,

0 0

where y is a column n-vector, is a row n-vector and aeR. The element Eego is given by

—1 0 0 E=( 000.

0 0 1

As in Example 4.1, we can describe the graded Lie algebra I as

I= V+ co(n)+ V*

under the identification

(0 'y 0 0 0 v)eg_ i —>yeV, 0 0 0

(0 0 0 0 0)egi —> eV*,

0 0

( a 0 0 0 A +90 —>A—ain eco(n). 0 0 a


Under this identification, the Lie algebra structure in 1 is given by

[y, y] =0, [, ']=O, [U, v]=U v, [, U]= U ,

[U, U']= U U' — U' U , [v,

where y, y'e V, c, ''e V*, U, U'e co(n). Clearly, the element E is now given by —Le co(n). Using the same bases for V, 11* and gl(n; R) as in Exam-ple 4.1, we can write the Maurer-Cartan form co of 0 (n +1, 1) as follows:

co=Ecoi ei +Eco.Vg +Ecoi ej,

where (coi) is co(n)-valued. Then the Maurer-Cartan structure equations of 0(n+ 1, 1) are given by

d coi = — E COI A COk ,

d wf = d wi = —E wk A wl.

Let P be a principal 20-bundle over M. Given a g_ 1 -valued 1-form co_ 1 and go-valued 1-form coo , we consider the question whether there exists a natural 91 -valued 1-form col such that co = co_ 1 + coo +col is a Cartan connection in P. There are some obvious conditions which must be imposed on co_ 1 and w0 . Corresponding to conditions (a), (b) and (c) for Cartan connections stated in § 3, we must have the following:

(a') co_ i (A*)= 0 and wo (A*)= A o for every A ego + gi ,

where A o is the go-component of A;

(b') W ar (co _ 1 + °k)= (ad a - 1 ) (w_ 1 + coo) for every a e 20 ,

where ad a-1 is the transformation of g_ 1 + go (= ligi ) induced by ad a': 1—> I;

(c') a tangent vector X of P is vertical (i.. e., tangent to a fibre) if co_ i (X)=0.

We are now in a position to state a theorem on normal projective and conformal connections.

Theorem 4.2. Let .2/ 20 be as in either Example 4.1 or Example 4.2 and P be a principal 20-bundle over a manifold M of dimension n (.. 3 for Ex. 4.2). Given a g_ 1 -valued 1-form w_ 1 =(cd) and a go-valued 1-form coo = (q) on P satisfying conditions (a), (V), (c') and

(I) doi i = — E wi A dc,


there is a unique Cartan connection co= co_ i +0).0 + col = (coi ; a4; coi) such that the curvature Q=(0; Cl .;; SO satisfies the following condition:

E 4, =o, where fl.;= E 1 K ii k i CO k A col .

This unique connection is called a normal projective or conformal connection according as 2/20 is as in Example 4.1 or 4.2.

Proof Let co=(coi ; coi; co]) be a Cartan connection with the given (a ; coi). In addition to the first structure equation (I) above, we have

(II)i, dcoi= —Ecoik A col— coi A coi + 6jEcoo co 14S4,

or

(II)c Ch.0ii= — E COL A (.0/jf — CO L A Wi — COi A (t)i± 6.1i DOk A COk ± qii ,

according as the connection is projective or conformal, and also

(III) dcoi = —Ewk A wi.; +Op

Applying exterior differentiation d to (I), making use of (I) and (II) and collecting the terms not involving col and wi, we obtain the first Bianchi identity:

QA CO-i =0, or equivalently,

14, 1 +K iku +Ki ik =0.

Hence, the condition EICjii =0 implies also

We shall now prove the uniqueness of a normal Cartan connection. Let Co =(coi ; coi; Q be another Cartan connection with the given (coi ; coi). By conditions (a) and (c) of §3, we can write

(T),—(0 .,= EAikC0k,

where the coefficients Aik are functions on P. Denoting the curvature of o by a=(0; 41 ; ai) and writing

6.= IL E kiki wk A COI,

we obtain by a straightforward computation using (II) the following relations between Ki m and kiki :

Poi — K.iiki= — bi A jk ± 451 A il+ (51.1Akl— Aik (projective),

kj k 1 — I C.j k 1 = — 61 AA+ 61 Ail+ M Aik — 6sit Ail+ 6. Akl — (5ijAlk

(conformal).


Hence,

(i)p E (kb,— ==(n + 1) (Ala Aik), (ii)p azi,„ 1) Ail+ (A11 — j

(i) EA„— (n — 2) Ail+ 6iiE A i ,,

(ii)c E = 2(n — 1) E Ali .

If both co and are normal Cartan connections, i.e., E Kj ii = E k.i„i = 0, then we see that A u = O and hence co= ro in either the projective or the conformal case. This proves the uniqueness of a normal Cartan connec-tion.

To prove the existence, assume first that there is a Cartan connection =(a.)'; c)j; coi) with the given (cd; col). We shall find a suitable system of

functions Aik so that ro =(cd; coi; C6 3) becomes a normal Cartan connec-tion. In the projective case, it is clear from (Op and (ii)p that it suffices to set

A1k= (n + 1)

1

(n — 1) E 4

n — 1 k

1 EKi ik .

In the conformal case, from (i) and (ii) we see that it suffices to set

1 1 A : k = 6 qc Kiiik .

j n-2 2(n-1) '

To complete the proof of the theorem, we have now only to prove that there is at least one Cartan connection co with the given (cd; 4. Let {U.} be a locally finite open cover of M with a partition of unity { . If co. is a Cartan connection in P IU,, with the given (cd; a4), then E (fx o 7E) COŒ

a is a Cartan connection in P with the given (cd; coi), where 1r: P— M is the projection. Hence, the problem is reduced to the case where P is a product bundle. Fixing a cross section a: M—> P, set co3(X)=0 for every vector X tangent to o- (M). If Y is an arbitrary tangent vector of P, we can write uniquely

Y Ra (X)+ W,

where X is a vector tangent to a(M), a is in 20 and W is a vertical vector. Extend Wto a unique fundamental vector field A* of P with A e!0 go + g. By conditions (a) and (b) for Cartan connections, we have to set

co(Y)— ad(a -1 )(co(X))+ A .

This defines the desired (coi). q.e.d.

(projective)

(conformal)


Theorem 4.3. Let P be as in Theorem 4.2 and co = (coi; coi; co) be a normal Cartan connection. Then

(1)E A coi =0, or equivalently, Kiki+ K iku+ K iuk= 0, where

Qi=E1-4 / cok A co';

(2)E

Qi = 0, or equivalently, Kiki+ Kku+ Kuk= 0, where

Q; =-EA-K iki w k A C01 ;

(3) if Qi= 0, then Q i = 0 provided dim M 3 in the projective case and dim M 4 in the conformal case.

Proof (1) This has been already proved in the proof of Theorem 4.2 whether the connection is normal or not as long as it satisfies (I) of Theorem 4.2. •

(2) Apply exterior differentiation d to (MI, and (II), collect those terms involving only w (not (.0.1 and co.) and take the trace. Since c/(21= 0, we obtain the desired result.

(3) Similarly, apply exterior differentiation d to (II) p and (II) and collect those terms involving only cd. Since (2.1=0 by assumption, we obtain

6m1 Kikt — brin Kai Kira1+ bic Ki m 45IK ikm+ (5I Kam = 0 .

Let i= m, and summing over i we obtain

(n— 3) Kikt+ (5i c E Kw + EKik,o. Summing over j= k, we obtain

(2n-4) EKiii =0.

From the last two identities we can conclude that K ik i- 0 provided n4. q. e. d.

Theorem 4.2 goes back to E. Cartan [2, 4]. Ochiai [3] proved Theo-rem 4.2 in a very general setting, thus showing that the existence and uniqueness of a normal Cartan connection are related to the vanishing of certain Spencer cohomology groups. Here we followed Kobayashi-Nagano [1] rather closely. For a slightly different approach, see Tanaka [1, 2]. See also Ogiue [1].

cos A Q. =0 (in the projective case),

co' A Qi — cal A Qi =0 (in the conformal case).

In the projective case, we can conclude immediately that Q. 0 provided n 3. In the conformal case, we rewrite the above identity as

5. Frames of Second Order 139

5. Frames of Second Order

In preparation for the following section on projective and conformal structures, we shall construct bundles of frames of higher order contact, in particular, of second order contact (see § 8 of Chapter I).

Let 'iti be an n-dimensional manifold. If U and I/ are two neighbor-hoods of the origin 0 of Rn, two mappings f: U-+ M and g: Ti—>M are said to defme the same r-jet at 0 if they have the same partial derivatives up to order r at O. The r-jet given by f is denoted by A(f). If f is a diffeo-morphism of a neighborhood of 0 onto an open subset of M, then the r-jet f0 (f) at 0 is called an r-frame at x = f (0). Clearly, a 1-frame is an ordinary linear frame. The set of r-frames of M, denoted by Pr(M), is a principal bundle over M with natural projection it, n(jro (f))= f (0), and with structure group Gr(n) which will be described next.

Let Gr(n) be the set of r-frames fo (g) at 0 ERA, where g is a diffeomor-phism from a neighborhood of 0 in Rn onto a neighborhood of 0 in W. Then Gr(n) is a group with multiplication defined by the composition of jets, i. e.,

iro (g) . j r0 (g1) = iro (g o g') .

The group Gr(n) acts on Pr(M) on the right by

A (f). j r0 (g) = j'0 ( f o g) for A (f )e Pr (M) and fo (g)e Gr(n).

Clearly, 131 (M) is the bundle of linear frames over M with group G 1 (n)= GL (n; R) .

From now on we shall consider only P1 (M) and P2 (M). If we consider the group 91(n; R) of affine transformations of Rn as a principal bundle over R" = 91(n; R)/GL(n; R) with structure group GL(n; R), we have a natural bundle isomorphism between 91(n; R) and the bundle Pi (R n) of linear frames over Rn:

91 (n ; R) 4--- Pi (R n)

I 1.

Under this isomorphism, the identity e of 21(n; R) corresponds to ,Po. (id), where id denotes the identity transformation of W. We shall therefore denote A (id) by e. The tangent space off.' (R") at e will be identified with that of 91(n; R) at e, that is, with the Lie algebra

a(n; R)=Rn+gi(n; R) of 9.1(n; R).


We shall now define a 1-form on P2 (M) with values in a (n; R). First, we observe that j1(f)—)j4(f) defines a bundle homomorphism 132 (M)—P1 (M). Let X be a vector tangent to P2 (M) at A(f) and X' be the image of X under the homomorphism P2 (M)—)131 (M). Then X' is a vector tangent to P1 (M) at j131 (f). Since f is a diffeomorphism of a neighborhood of the origin 0 of Rn onto a neighborhood off(0)eM, it induces a diffeo-morphism of a neighborhood of eeP1 (IV) onto a neighborhood of j4(f)eP 1 (M). The latter induces an isomorphism of the tangent space a (n; R)= 7; (P1 (Rn)) onto the tangent space of P1 (M) at j4(f ); this iso-morphism will be denoted by! and easily seen to depend only on A(f). The canonical form 0 on P2 (M) is an a (n; R)-valued 1-form defined by

0 (X) = f -I (X') .

The construction generalizes that of the canonical form of 131 (M). We define the adjoint action ad of G 2 (n) on a (n; R) as follows. Let

A (g)e G 2 (n) and A(DeP 1 (Rn). The mapping of a neighborhood of e E P1 (W) onto a neighborhood of e E P1 (W) defined by

A (f )—)ii(1) (g of o g— 1 )

induces a linear isomorphism of the tangent space a (n; R)= 7e (P1 (m) onto itself. This linear automorphism of a (n; R) depends only on j (g) and will be denoted by ad (j (g)).

Since G 2 (n) acts on P2 (M) on the right, every element A of the Lie algebra g2 (n) of G2 (n) induces a vector field A* on P2 (M), called the fundamental vector field corresponding to A

Proposition 5.1. Let 0 be the canonical form on P2 (M) defined above. Then

(1) 0(A*)= A' for A E g2 (n), where A' e gl (n)= gl (n ; R) is the image of A under the. homomorphism g2 (n)—> g1 (n);

(2) (R.)* 9=: ad (a -1 )0 for a e G 2 (n). The proof is straightforward. , We shall now express the canonical form of P2 (M) in terms of the

local coordinate system of P2 (M) which arises in a natural way from a local coordinate system of M. For this purpose we may restrict ourselves to the case M = W. Let e1 , ... , en be the natural basis for Rn and (x1 , ... , xn) be the natural coordinate system in W. Each 2-frame u of Rn has a unique polynomial representation u=j(f) of the form

f (x)=E(u s +Eu 1 xi +4Eulk xi xk) ei ,

where x =E x1 ei and 14 1, -=uli . We take (u1 ; 14; ulk) as the natural coordinate system in P2 (Rn). Restricting (i4; f.4 k) to G 2 (n) we obtain the

6. Projective and Conformal Structures 141.

natural coordinate system in G 2 (n), which will be denoted by (si; The action of G 2 (n) on P2 (W) is then given by

(ui ; uS; ul k)(sii ; 4 k)=(ui ; uip sy ; E upi syk +E uqf r

In particular, the multiplication in G 2 (n) is given by

Similarly, we can introduce the natural coordinate system (ui ; u. ) in P1 (W) and the natural coordinate system (.s .ii) in G 1 (n) so that the homo-morphisms P2 (R n) —> P1

(Rn) and G 2 (n)—> (n) are given by

(ui ; (zit ; uti) and (si; sik) —>

respectively. Let (Ei ; El} be the basis for a (n; R) defined by

and set Ei = la , El= (i/a di) e

0=E0t E1 -FE0jEj.

From the definition of the canonical form 0 and the action of G 2 (n) on p2 (Ra) expressed in terms of the natural coordinate systems, we obtain the following formulae:

dui =Eulei,

duti —Eulei+Euihi 0h.

Let (vi) be the inverse matrix of (i4). Then

O 1 —>v duk, =Evik

From these formulae we obtain the following structure equation:

Proposition 5.2. Let 0 = (0 i ; O. ) be the canonical form on P 2 (M). Then

d0 1 =—E0ik nOk .

For the canonical form on P' (M) for r >2 and its structure equation, see Kobayashi [8].

This set-up of jets, frames of higher order contact, ete. is due to Ehresmann [3].

6. Projective and Conformal Structures

Let 2/20 be as in Example 4.1 or 4.2. We can consider .2 0 as a subgroup of the group G2 (n)-- ((c ) ; aiik)) defined in §5 as follows. Let o denote the origin of the homogeneous space 2/2 0 and consider each element g of


20 as a transformation of 2/2 0 leaving the origin o fixed. It can be easily verified that j(23 (g)=id if and only if g = id, that is, every element of 20 is determined by its partial derivatives of order 1 and 2 at the origin o. Hence, 20 is isomorphic to the group of 2-jets (A(g); g ego). Choosing a basis for g_ i , identify g_ i with V= Rn as in Example 4.1 or 4.2. Then the mapping

Rn = 1 e'>' 2—> 2/20

gives a diffeomorphism from a neighborhood of 0 eRn onto a neighbor-hood of o E 2/20 and defines a local coordinate system around o ewe ° . With respect to this coordinate system, each 2-jet A(g) is an element of the group G 2 (n). Hence, 20 can be considered as a subgroup of G 2 (n). An explicit description of 2 0 as a subgroup of G2 (n) is not without interest although it is not essential in the subsequent discussion. Let N2 (n)= MO) denote the kernel of the natural homomorphism G 2 (n)—> On) which sends (ai; ciik) into (ai). In view of the diagram of exact sequences:

—> (n)—> G 2 (n)—> (n) —> 1

U U U 0—> 21 —> 20 —> 20/21 —> 1,

we shall describe 2 0/21 as a subgroup of G 1 (n) and 21 as a subgroup of N2 (n).

First, let weo be as in Example 4.1. Then 20/21 = G1 (n)= GL(n; R) and 21 = {(dik);

Next, let 2/20 be as in Example 4.2. Then 2 0/21 = CO (n) and

21 = ((aik); a.ik= ± J i) •

Thus 21 coincides with the first prolongation of the Lie algebra co(n) (see Example 2.6 of Chapter I).

Let M be a manifold of dimension n and P 2 (M) be the bundle of 2- frames over M with structure group G 2 (n) (see § 5). A principal subbundle P of P2 (M) with structure group 20 (c G2 (n)) is called a projective structure or a conformal structure on M according as 2/20 is as in Ex-ample 4.1 or 4.2. (In Example 2.6 of Chapter I, a CO(n)-structure on M was called a conformal structure on M. The two definitions are equi-valent in the following sense. If P is a conformal structure as a subbundle of P2 (M), then P/2 1 is a subbundle of P 1 (M).- L(M) with group 20/21 = CO(n) in a natural manner and hence is a CO(n)-structure. Conversely, if Q is a CO (n)-structure on M, then its first prolongation Q1 (see §5 of Chapter I) is a conformal structure as a subbundle of P 2 (M). This gives a one-to-one correspondence between the conformal structures P c P2 (NI) and the CO (n)-structures Q c P 1 (M). Since we shall not use this fact, the

6. Projective and Conformal Structures 143

proof is left to the reader. But a projective structure P cannot be defined as a subbundle of P1 (M)= L(M) since P/21 is 131 (M) itself and the first prolongation of P1 (M) is precisely P2 (M).)

Now, let P c P2 (M) be a projective or conformal structure on M and (cot ; co be the restriction to P of the canonical form (O'; Of,) of P 2 (M) (see § 5). By Propositions 5.1 and 5.2, (co e ; cf4) satisfies the assumptions (a'), (b), (c') and (I) of Theorem 4.2. We shall therefore call (coi; 04) the canonical form of P. By Theorem 4.2, there is a unique normal projective or conformal connection (co 1 ; co. ; co) provided n 3.

Let P and P' be projective (resp. conformal) structures on manifolds M and M', respectively. A (local) diffeomorphism f of M into M' induces a local isomorphism f* of the bundle P 2 (M) of 2-frames of M into the bundle P2 (M') of 2-frames of M'. If f* sends P into P', then f is called a (local) projective (resp. conformal) isomorphism of M into M'. If M = M' and P = P', then a projective (resp. conformal) isomorphism f is called a projective (resp. conformal) transformation or automorphism. This defini-tion is completely analogous to that of an automorphism of a G-structure given in § 1 of Chapter I. An infinitesimal projective or conformal trans-formation can be defined in the same way as an infinitesimal automor-phism of a G-structure.

Assume that n 3 so that the normal projective or conformal connec-tion is unique. Then, for each automorphism f of P, f (restricted to P) preserves the normal connection co =(w'; coii ; coi). From Theorem 3.1 we obtain (Kobayashi [2]).

Theorem 6.1. If P is a projective or conformal structure on a manifold M of dimension n 3, then the group Qt of projective or conformal transfor-mations is a Lie transformation group of dimension dim P (= n2 +3 n in the projective case and = + 1)(n +2) in the conformal case).

In order to determine the cases where dim 91 , dim P, we consider some examples.

Let 2/20 =I(R) be as in Example 4.1, where .2 = PGL(n; R). The principal bundle .2 over 2/20 with group 20 can be identified with a projective structure on 2/20 in a natural manner. To describe this identification, let o denote the origin (i.e., the coset 2 0) of the homo-geneous space 2/20 . Since each fe .2 is a transformation of 2/20 and a neighborhood of o in 2/20 is identified with a neighborhood of 0 in R n in a natural way, the 2-jet j (f) can be considered as a 2-frame of 2/20

atf(o). The set of all 2-frames thus obtained defines a projective structure on 2/20 which can be identified with the bundle L over 2/20 . Then the Maurer-Cartan form co =(co1 ; coi ; ao of .2 defined in Example 4.1 defines the normal projective connection of this projective structure. The Maurer-Cartan structure equations of .2 show that the connection has


no curvature. Clearly, Q is the group of projective transformations of this projective structure.

We consider now the universal covering space sr: of F(R). Since the covering projection I(R) is a local diffeomorphism, the natural projective structure on I(R) described above induces a projective structure on Sn. We shall give a group-theoretic description of this natural projective structure on S". Let .2 = GL(n + 1; R)/R+, where R + denotes the normal subgroup of GL(n +1; R) consisting of elements cdn+i with a> O. It is a group with two components which is locally isomorphic to PLG(n; R). (PLG(n; R)= GL(n + 1; R)/R*, where R* is the normal subgroup consisting of elements a I n +1 with a+ 0.) Let

20=1 (A eGL(n+1; R); a>01/R+, a

where A E GL(n ; R) and is a row n-vector. Then Sn =2/20 . The principal bundle 2 over 2/20 with group 20 is the desired projective structure on Sn which is locally isomorphic to the natural projective structure on pn (R) under the covering projection Sn—)1(R). The group of projective trans-formations of this projective structure is Q.

As the third example, let 2/20 sr, be as in Example 4.2, where 2= O (n +1, 1). Then the principal bundle 2 over 2/20 can be naturally identified with a conformal structure on 2/20 . Again the Maurer-Cartan form co= (a); coi; co) of Q defined in Example 4.2 defines the normal con-formal connection of this conformal structure. As we can see from the Maurer-Cartan structure equations of 2, the connection has no curvature. The group .2 is precisely the group of conformal transformations of this conformal structure.

Theorem 6.2. In Theorem 6.1, assume dim 91= dim P. In the projective case, P is the natural projective structure on either Pn (R) or its universal covering space Sn as explained above. In the conformal case, P is the natural conformal structure on Sn described above.

We shall only indicate the main idea of the proof. We consider the projective case, the conformal case being similar. Since each 1-parameter group of projective transformations of M lifts to al-parameter group of projective transformations of the universal covering space JÇ , the group of projective transformations of /i/-1 has the maximum dimension dim P. We shall first assume that M is simply connected. Choose.a point u0 of P. We know (Theorem 3.2 of Chapter I) that the mapping fe91—>fit (uo)eP is injective and the image is a closed submanifold of P which can be identified with 91 (as a differentiable manifold). Since dim 91= dim P by assumption, the identity component of 91 can be identified with one of the components of P under the mapping 91—) P defined above. (Since P

7. Projective and Conformal Equivalences 145

has two connected components, 91 can be identified with P if 91 is, not connected.) Let xo be the base point of uo in M and 910 be the isotropy subgroup of 91 at xo . Since 91 is fibre-transitive on P, it is transitive on M so that M = 91/91 g . Let co = (d; cc4; wi) be the normal projective connec-tion of P. Since the forms (a); q; co) are invariant by 91, restricted to 91c P they are Maurer-Cartan forms of the group 91. Since the curvature of co vanishes by Theorem 3.2, the structure equations (I), (II)p , (III) of § 5 are nothing but the Maurer-Cartan structure equations of the group 91. From these structure equations we see that 91 is locally isomorphic to 2 = GL(n+ 1; R)/R +. The identity component of 210 coincides with the identity component of the structure group 20 . Since both 91/210 and 2/20 are simply connected, the standard argument proves that the identity component of 21 and the identity component of 2 are isomorphic to each other not only as groups but also as bundles over M. 91/910 and Sn =2/20 respectively. Then it follows that the bundle P over M is isomorphic to the bundle 2 over Sn. If M is not simply connected, then M= Sn/T, where T is a discrete subgroup of 2 = GL(n +1; R)/R + which commutes elementwise with the identity component GL E (n + 1; R)/R + of 2 as in the proof of Theorem 3.1 of Chapter II. (Here GL + (n +1; R) denotes the identity component of GL(n +1; R), which consists of matrices with positive determinant.) Then we see that 11 consists of two elements represented by + i n+i , and that P is the natural projective structure on M = pn (R).

7. Projective and Conformal Equivalences

We shall first explain projective equivalence of affine connections. Let I= g_ 1 + go + git = Y+ g! (n; R)+ 17* be as in Example 4.1. Let P1 (M) (= L(M)) be the bundle of linear frames over M and 0 =WI be the canonical form of P1 (M), viewed as a g_ 1 -valued 1-form as well as a Y-valued 1-form. Two torsionfree affine connections of M defined by go-valued 1-forms co.(coi) and co' =((o'i ) are said to be projectively equi-valent if there exists a grvalued function p = (pi) on P1 (M) such that

co' — co= [0, p].

This formulation is due to Tanaka [4] (see also Kobayashi-Ochiai [1]). Note that the left hand side takes values in go and the right hand side in [g_ 1 , gi ] c go . If we consider co and co' as gl(n; R)-valued forms, 0 as a Y-valued form (i.e., a form whose values are column n-vectors) and p as a Y*-valued function (i.e., a function whose values are row n-vectors), then the relation above may be rewritten as follows:

co' —0)=-Op+(p0)1„,


or more explicitly

19' P (E °I‘ P k) If we set

then the relation reads as follows:

ri k =61pi +5ipk .

Similarly, we define conformal equivalence of affine connections. Let I= g_ 1 + go + git = V+ co (n)+ V* be as in Example 4.2. Let Qc P1 (M) be a CO (n)-structure over M and O = (0) be the canonical form on Q. Two torsionfree connections in Q defined by go-valued 1-forms co = and co' =(ctii i) on Q are said to be conformally equivalent if there exists a g1 -valued function p =(p i) on Q such that

cd —0)=M A,

or, in terms of matrix and vector notations,

al —0)=0 p—'pl0+(p19)

or more explicitly,

— coi = 19' p — 9i p,+ (E O k p

If we set wJ -wJ=rJk O,

then the relation reads as follows:

ilk= 61P biPi+

Although two torsionfree connections in P1 (M) are not necessarily pro-jectively equivalent, two torsionfree connections c o = (0)i) and co'=(ali i) in the CO(n)-structure Q are always conformally equivalent to each other. This difference comes from the fact that while' g 1 is the first pro-longation of go in the conformal case, g1 is strictly smaller than the first prolongation of go in the projective case. In fact, at each point of Q, (ri k) defines an element of the first prolongation of go since, for each fixed k, the matrix (riik)0.0i , ...,„ is in co(n) and (r.k) is symmetric in j and k. In Example 2.6 of Chapter I, we proved that each element (ilk) of the first prolongation of co (n) determines a unique vector (pi) such that rt., k = 61 biPi+ (5; Pk'

In order to relate the notion of projective equivalence to that of projective structure, we prove the following


Proposition 7.1. Let P2 (M) be the bundle of 2-frames over an n-dimen-

as the subgroup of G 2 (n) consisting of elements (si; s) with sik =0 in terms sional manifold M with structure group G 2 (n). Consider G 1 (n)=--GL(n; R)

of the natural coordinate system introduced in § 5. Let 2/ 20 . g(R) be as in Example 4.1 and consider .20 as a subgroup of G 2 (n) as in §6 so that

(n) 0 c G 2 (n). Then

(1) The cross sections M— P2 (M)/G 1 (n) are in one-to-one correspond-ence with the torsionfree affine connections of M.

(2) The cross sections M—> P2 (M)/ 130 are in one-to-one correspondence with the projective structures of M.

Proof (1) Let (u'; 4) be the natural local coordinate system in (M) induced from a local coordinate system x 1, , xn of M as in § 5.

We introduce a local coordinate system (zi; zi k ) in P2 (M)/G 1 (n) in such a way that the natural projection P 2 (M)—> P2 (M)/G 1 (n) is given by the equations zi= = uipq fic where

(v.)

.:(0-1 (the inverse matrix).

Then a cross section : M —> (M)/ (n) is given locally by a set of functions

risk (xl , , xn) with ri -Ti jk kj •

If we consider the action of the group G2 (n) on the fibre G 2 (n)/G 1 (n), then we see that the functions ilk behave under the coordinate changes as Christoffel's symbols should. This proves (1).

(2) Since the reductions of the structure group G 2 (n) to 20 are in one-to-one correspondence with the cross sections M —> P2 (M)/ 2 0 , (2) is evident. q. e.d.

Let 0 =(13 : • 0) be the canonical form on P 2 (M) defined in §5 and y: 131 (M) c--> (M) be the reduction of the structure group G 2 (n) to G 1 (n) corresponding to a cross section : M — > P 2 (M)/ (n). Then (y* (01) is the canonical form of 131 (M) and (y*(60i)) is the connection form cor-responding to F. This follows easily from the proof of Proposition 7.1 and the expression of 0 given in §5 in terms of the coordinate system (u`;

Every torsionfree affine connection F: M—* P2 (M)/G 1 (n), composed with the natural mapping P2 (M)/ G 1 (n)—> P 2 (M)/20 , gives a projective structure M —> P 2 (M)/2 0 . A torsionfree connection r is said to belong to a projective structure P if it induces P in the way described above. •

Proposition 7.2. Two torsionfree affine connections of a manifold M are projectively equivalent if and only if they belong to the same projective structure.

148 IV. Milne, Conformal and Projective Transformations

Proof Let I' and I" be two cross sections M —) 132 (M)/G 1 (n), i.e., torsionfree affine connections and let co and co' be the corresponding connection forms on 131 (M). A straightforward calculation shows that co and co' are projectively equivalent if and only if there is a 1-form p= p i dx' on M such that

Gcg — Elk = (51 Pk -F (51P],

where Tiki and Fik are Christoffel's symbols for co and co' with respect to a local coordinate system , xn. This in turn is equivalent to the condition that I" and I" induce the same cross section M—> P2 (M)/2 0 , because the kernel 21 of the homomorphism Q 0 —> On) consists of elements (ah) of the form a.k = pk + 5 p (see § 6). q. ed.

Given a torsionfree affine connection on M, let 91(M) be the group of affine transformations of M and 13(M) be the group of projective trans-formations, i. e., automorphisms of the induced projective structure. Proposition 7.2 implies that a transformation of M is a projective trans-formation if and only if it transforms the given connection into a pro-jectively equivalent affine connection. Evidently, we have the inclusion 91(M) c 43 (M). There are cases where these groups have the same identity component. We quote only the following result of Nagano [8]. .

Theorem 7.3. Let M be a complete Riemannian manifold with parallel Ricci tensor. Then the largest connected group 130 (W) of projective trans-formations of M coincides with the largest connected group 9.1° (M) of affine transformations of M unless M is a space of positive constant cur-vature.

For other related results, see Couty [2, 3], Ishihara [2, 3], Solodov-nikov [1], Tanaka [2], Tashiro [3], Yano [3], Yano-Nagano [2], and references therein.

Given a CO (n)-structure Q c P1 (M), let P c P2 (M) be the correspond-ing conformal structure on M. Then a transformation of M is an auto-morphism of the CO (n)-structure Q if and only if it is an automorphism of the conformal structure P. Such a transformation is called a conformal transformation of M. If M is a Riemannian manifold, then a transfor-mation of M is a conformal transformation with - respect to the naturally induced CO (n)-structure if and only if it sends the metric into a con-formally equivalent metric (cf. Example 2.6 of Chapter I). Evidently, the group E (M) of conformal transformations of M contains the group 3(M) of isometries of M. We quote only two results on the relationship between E (M) and 3(M).

Theorem 7.4. Let M be a complete Riemannian manifold with parallel Ricci tensor. Then the largest connected group E° (M) of conformal trans-


formations of M coincides with the largest connected group 3° (M) of isometries of M unless M is isometric to a simply connected space of positive constant curvature ( i.e., a sphere).

This result of Nagano [1] extends the result of Yano -Nagano [1] on Einstein manifolds. The following theorem was proved by Obata [8]. See Lelong-Ferrand [4] for related results.

'theorem 7.5. Let M be a compact Riemannian manifold with constant scalar curvature. Then E° (M) . 3° (M) unless M is isometric to a simply connected space of positive constant curvature.

For other related results, see Goldberg-Kobayashi [1], Kulkarni [1], Ledger-Obata [1], Lichnerowicz [1], Obata [3-7, 9], Suyama-Tsuka-moto [1], Tanaka [1], Tashiro [2], Tashiro-Miyashita [1, 2], Weber-Goldberg [1], Yano [3, 8], Yano-Nagano [2], Yano-Obata [1], and references therein. A list of papers which had given partial results toward Theorem 7.5 can be found in Yano-Obata [1].

Appendices

1. Reductions of 1-Forms and Closed 2-Forms

For the discussion of contact structures and symplectic structures, it is important to know the simplest possible expressions for 1-forms and closed 2-forms.

Let a) be a 1-form defined on a manifold M of dimension n. We set d2k) = CO A ••• A da) (k times),

(2k+I)= condom— Ada) (da): k times).

Then cdP ) is a form of degree p. We say that a 1-form a) is of rank r at a point o if co(r) +0 but aP+ 1) =0 at o. Then we have

Theorem 1. If a 1-form a) is of rank p in a neighborhood of a point o, then there exists a local coordinate system x1 , xn around o such that

(0=x ' dx2 + x3 d x4+ ± x2k-1 d x2k for p = 2 k,

w=x dx2+x3 dx4+• +x2k-1 dx2k+dx2k+i for p=2k+1.

Proof We first prove

Lemma 1. If co is of rank p, then there exist a neighborhood Un of o, a fibring Un—> UP with dim UP =p and a 1-form Co on UP such that ir* (&)=w. In other words, with a suitable choice of local coordinate system, co depends only on the first p coordinate functions.

Proof of Lemma 1. At each point we consider the space S of tangent vectors X such that

ix a)=0 and tx da)=0,

where ix denotes the interior product. It is straightforward to verify that if X and Y are vector fields which belong to S at each point, then [X, Y] belongs to S at each point. We shall show that dim S=n—p. (That will mean that S defines an involutive distribution of dimension n — p.) Set p =2k or p =2k +1 according as p is even or odd. Then at each point,

1. Reductions of 1-Forms and Closed 2-Forms 151

da) is a skew-symmetric bilinear form of rank 2k on the tangent space. If we denote by S' the space of tangent vectors X such that

ix da)=0,

then dim S' = n —2k and S c S'. If p= 2k, then S = S'. In fact, if S *S' and YES' is an element not in S, then co(Y)* 0. If X1 , ... , X2 k are tangent vectors (not in S') such that

V 2k1 1 = 010) A • • • A d(o)(X l , ..., X 2 0#0, Ji L

then co(2k+1)11, , ki, A I , ..., X2k)=00 A dWA ••• A dC0)(Y, XI , ..., X20

=COOT a (2k) (XI, ...,X20#0,

in contradiction to the assumption that cook" = 0. If p= 2 k+ 1, then dim S' =1+ dim S. In fact, since S = {X e S'; a)(X)= 0), we have dim S S dim S' 1+ dim S. It suffices therefore to prove S* S'. Since w(2k + 1) *0, there exists a (2k + 1)-dimensional subspace T of the tangent space at each point such that the restriction of aP k + 1) to T is nonzero. In other words, for any basis X11 ...,X2k +1 of T, we have a 2k + 1) (XI , ..., X 2k+1 )# O. But, if S=S', then

dim(Sn T)dimS+dim T—n=n-2k+2k+l—n=1.

By taking a basis X i , ... , X2k +i of T in such a way that X1 ES nT, we obtain

(0(2k+l)ty y k.exi, ...m2, ••• 1 X2k + i)=(.0 A dw A ••• A dco)(X i , X2, ••., X2k + i)

=0

since ixi co =0 and i1 da)= O. This is obviously a contradiction. We have shown that S defines an involutive distribution of dimension

n — p. Consider the maximal integral submanifolds defined by the distribution. They give a fibring of a suitable neighborhood Un of o whose fibres are of dimension n — p. We denote this fibring of Un by n: U's— > U. From the definition of S it is clear that co can be projected onto UP, that is, there exists a (unique) 1-form 6 on UP such that n* (6' )= co. This completes the proof of Lemma 1.

Lemma 2. If a) is of rank n, then there exists a function f such that df *0 and df A dn-1) =0 in a neighborhood of o.

Proof of Lemma 2. We write

an - 1) .E (— oi ai dX1 A •• • A dx i A • •• A dxn. i

152 Appendices

Then the equation df A co(n-1) = 0 reduces to

.o.

f a xi

This partial differential equation admits (infinitely many) solutions f such that df *0 in a neighborhood of o.

Lemma 3. If co is of rank p, then there exists a function f such that df *0 and dfn

Proof of Lemma 3. This is immediate from Lemmas I and 2.

It is also evident that in Lemma 3 we can choose f in such a way that it is constant on each fibre in the fibering Un P given in Lemma 1.

We say in general that a form co on M depends on parameters ti , , tm if it is a form on M x N, where N is the space of parameters such that its restriction to (p) x N vanishes for every point p of M, i.e., such that it does not involve d t 1 , , d t„, when expressed in a local coordinate system. The exterior derivative dco of such a form co on M which depends on the parameters t1 , , tm is taken considering , tm as constants. In other words, we take the ordinary exterior derivative of co as a form on M xN and then delete the terms containing dt i ,...,dt m . Looking at the proofs of Lemmas 1 and 2, it is clear that Lemma 3 may be generalized to the case where co depends on parameters. We have

Lemma 4. If, in Lemma 3, co depends on m parameters t1 , , t„„ then we can find a function f which depends differentiably on , tm .

Making use of these lemmas, we shall now prove the theorem. We denote by y' the function f obtained in Lemma 3 and try to find a new function f such that

dfn d yl * 0 and dfn d y l co(P - 3) =0.

Choose a local coordinate system y', u 2 , , un around o; since d y l *0, such a local coordinate system exists. We write

w=hdy l +q),

where q) does not contain dyi. We consider a neighborhood of o of the form C.J1 X U2 such that y' is a coordinate system in LI, and u2, , Un is a coordinate system in 112 in a natural manner. Then q) may be considered as a 1-form on U2 which depends on the parameter ;I'. We shall denote (p by ep when we consider it as a form U2 which depends on the parameter y'. By direct calculation we obtain

dy l A coo = d ep'(J) for every j 1.

1. Reductions of 1-Forms and Closed 2-Forms 153

Setting j = p —1, we see that ip (P - 1) =0. By direct calculation we obtain also

= (k a d + de()) A 2),

where p = 2k or p= 2 k + 1 according as p is even or odd and a is a 1-form such that dco =a A dy l +chp. Since co(P) *0, we have ep" (P- 2) 40. Hence ip is of rank p — 2. Applying Lemma 4 to rp, we obtain a function f on U2 such that df *0 and df A 45( =0. We can extend f to a function on Ui x U2

in a natural manner; the extended function will be denoted also by f Then dyl A df * 0 and d df A CP ." 3) = d yl A df A rp (P - 3) = 0. Setting y2 =f, we may write

d d y 2 + 0 and dyl A dy2 Aco(P -3) =0.

Continuing in this way we obtain functions yic such that

d A • • • A d and

d A ... A d A co(P- 2k + 1) =0

According as p = 2 k or p = 2 k + 1, we have

dy l ••• dy k co=0 if p=2k,

d Ady k Adco=0 if p=2k+1.

If p = 2 k, we have therefore co=zi d yl ± z2 d y2 ±zk d y k ,

where z1 , zk are functions. Since OP = d co A ... A d co 4 0, it follows that d 3/1 A .•• A d y k A dz 1 A ... A d z k 4 0. Setting

X1 =Z1 , x2=y1,...,x 21c-1 = Zk X 2 k =yk

and defining x 2 " 1 , , xn suitably, we obtain a local coordinate system , xi' with the desired property. If p = 2 k +1, we take a local coor-

, ,

yl y k uk+1 , ... , u dinate system n ; since d ••• A d yk + 0, such a coordinate system exists. We then write

dy 1 +.•.+hk dy k +0,

where is a 1-form which does not involve dy l , , dy k. As we have explained before Lemma 4, ik may be considered as a 1-form on the space of uk + 1 , tin which depends on the parameter , yk and will be denoted by when it is considered as a form in uk + 1 , un. Hence dik is obtained from dtp deleting the terms involving dy i , ...,dy k . Since 0= dy l A ... A dy k A dco=dy l A ••• A dy k Ad tk, it follows that d =0. By

yl , + 1 Poincar6 lemma, there exists a function g of y k, uk un such

154

Appendices

that dg=ili+gl dy 1 +•••+gk dy k,

where gi = a ea y f. Hence we have

co=(hi —g1 ) dy 1 +•••+(hk —gk)dy k +dg.

If we set xl = —gi, 2x =y1 , x2k--1 = hk _ x2k = yk, x2k+1 = _ , g then dx 1 A dX 2 A ••• dx 2k + 1 +0 since a 2k + 1) +0. Choosing x 2k + 2 , ...,xn suitably, we obtain a local coordinate system with the desired pro-perty. q. e.d.

Let 0 be a 2-form defined on a manifold M of dimension n. We say that 0 is of rank 2p at a point o if Q =QA A (p times) is nonzero but 02 + 1 =0 at o. We say that Q is of maximal rank if it is of rank n. As an application of Theorem 3.1, we prove

Theorem 2. If a closed 2-form Q is of rank 2p in a neighborhood of a point o, then there exists a local coordinate system x',...,xn around o such that

w=xidx2 +...+x2 P-1 dx2 P Or

co=xidx2 +.-±x2 P-1 dx 2 P+dx2 P+1 .

Then it is evident that Q has a desired expression. q.e.d.

Theorem 3.1 is due to Frobenius and Darboux. We followed the presentation of E. Cartan in [9] to which we refer the reader for relevant references. See also Arens [2].

2. Some Integral Formulas

We prove first the following integral formula of Yano [6] (see also Yano and Bochner [1]).

Theorem I. Let M be a compact, orientable Riemannian manifold with Riemannian connection V and Ricci tensor S. Then, for every vector field X on M, we have

S (S(X, X)+ trace(Ax Ax)—(div X) 2) d v =0 ,

C2=dx 1 dx 2 +...+dx 2 P -1 Adx.

Proof. Since Q is closed, by Poincaré lemma we have Q=dco, where co is a 1-form defined in a neighborhood of o. Then co is of rank either 2p or 2p +1. By Theorem 3.1 there exists a local coordinate system x 1 , xn around o such that either

2. Some Integral Formulas 155

where Ax is the field of linear endomorphisms defined by Ax Y = —Vy X, div X is the divergence of X and dv denotes the volume element of M.

Proof The proof is in terms of local coordinates. It suffices to express the integrand as the divergence of a vector field. Let e be the components of X with respect to a local coordinate system x', , xi'. Then

div (A x X)= —E e

-E • + • v

-E (vi • e+ RIO +Vj e • Vi i,j

= e • ±Vj • VI V) i,j

and •

=E vi e • V+ • NW).

From these two equalities, it is clear that the integrand in Theorem 1 is equal to — div (A x X) — div ((div X) X). q. e. d.

In terms of a local coordinate system, the formula in Theorem 1 reads as follows:

E (Ri; e +Vi e • V —V; • Vi 0=0. f,

From Theorem 1, we obtain immediately the following three formulas:

Corollary. Hith the same notations as in Theorem 1, we have

(1) J {S(X, X)+ trace(Ax Ax)-Ei trace ((Ax — A x)2)— (div X) 2} dv=0;

(2) f IS(X, X)— trace (A x .14x)—i trace ((Ax + A)2)— (div X) 2 } dv=0;

f IS(X, X) — trace (A x x)

+4 trace (( Ax +14x — —2 (div X) .02) — n — 2 (div X)2} dv=0. (3)

We shall now prove a Kdhlerian analog of Theorem 1.

Theorem 2. Let M be a compact Kahler manifold with Ricci tensor S. For any complex vector fields X and Y of type (1,0), we have

{S (x, r7)+ trace(A;:. 4)— (div X)(div V)) dv=0 ,

156 Appendices

where A x" is the field of linear transformations sending a vector field of type (0, 1) into the vector field —V E X of type (1, 0), i.e.,

—VE X.

Before we start the proof, we remark that A sends a vector field of type (1,0) into a vector field of type (0, 1) so that Ax" • diiT sends a vector field of type (1,0) into a vector field of the same type. We note also that, by setting X = y in the formula above, we obtain a Kahlerian analog of the formula in Theorem 1.

Proof In terms of a local coordinate system zl, zn, let ta a a a

x = and Y=Etr .

a za a zet

Then the components of Ai; are given by — The components of the vector field A(Y)= — Vy X of type (1,0) are given by —E ?. A calculation similar to the one in the proof of Theorem 1 yields

—E vcc (vpdg TIP). — (V, va • TIP + Œ ill + • )

E yva•iP)=E(vp ± Vac a Vg

By adding these two equalities and integrating the resulting equality over M, we obtain

E (Rap OE -r-71 1 + Vij • VŒ F —VŒ Œ • V p ill) dv =0 , Af

which is precisely the formula in Theorem 2. q.e.d.

Let X be a vector field on a Riemannian manifold M. Let be the 1-form corresponding to X under the duality defined by the Riemannian metric. If we denote by A the Laplacian, then A c is a 1-form. We denote by A X the vector field corresponding to A We state

Theorem 3. Let M be a compact orientable Riemannian manifold with Ricci tensor S. Then, for any vector fields X and Y on M, we have

(— (A X, Y)+ S(X, Y)+ (VX, VY)) dv=0. At

Proof Let c and ni be the components of X and Y with respect to a local coordinate system x', , xn. Combining

E co .10=E vi ni +v' Nini) with the formula for the Laplacian (see Appendix 3), we obtain

Evvi • to= —DA V. tli+ERuVri i +EVT*Virli.

This, integrated over M, yields the desired formula q.e.d.

and

3. Laplacians in Local Coordinates 157

Let X be a complex vector field of type (1,0) on a Kdhler manifold Ai a

and let be the corresponding (0, 1)-form. (If X = —, then a za = E dr, where = E v3.) Let A" =6" d" + d" 6" be the d"-Lapla-

cian. Then A" c is a (0, 1)-form. We denote by 4" X the corresponding vector field of type (1, 0). As a Kdhlerian analog of Theorem 3, we obtain

Theorem 4. Let M be a compact Kahler manifold with Ricci tensor S. For any complex vector fields X and Y of type (1, 0), we have

(— (A" X, Y)+ S(X, Tr) + (v" x, V" Y)) dv = 0.

Proof. From

Eva(va e • t70)=-E(va va e3 • Fil+ va v3 • vcc and from the formula for the d"-Laplacian (see Appendix 3), we obtain the desired formula in the same way as in the proof of Theorem 3. q.e.d.

Expressed in terms of local coordinates, we obtain

{—E(z1"Dpie-FERzpŒijfl+EV0 Œ.Vs zt} dv=0.

3. Laplacians in Local Coordinates

C4)=-F EW1112...ip dx il A dxi2 A A dx iP

be a local expression for co. Then

1

dco= 4 . 4 ••• lp (p+1)!

dx A dxfl A ... A dx fP, and

1

co = . dx f2 A dxiP. (p-1)! . 12 ...1„

In particular, let co be a 1-form so that

co=Ecoi dx i.

da)=1E(Vi coi —Vi co1)dx 1 ndx1,

dco= —EVÎ (V i coi —Vi coi)d

bco=

dbco= —DOW dxj.

Let co be a p-form on a Riemannian manifold M. Let

1

Then

158 Appendices

Hence, do=bdco+dbco

=E( — NW' wi+ViVid —V j Vi coi)d xi

=E( —DI V coi + R t., col) d x- I .

For a similar expression for a p-form co, see Yano [9; p. 67]. Let co be a (p, q)-form on a Kahler manifold M. Let

1 v, co = w d Zœl A • • • A d ZIP A d el A - • A d 21341 p ! q ! Lai ... ap Pi ••• /14

be a local expression for co. Then

1 q d" a) = p! (q+1)! E (vii c°44 Pi ••• Jig — kEi VPk (DA A ... 4-1 Pok + 1... ff.)

. d z A A d2P A dgl A — A deg,

where A stands for oc1 ... al, and d z A for dza 1 A • • • A deg. We have

1 VE 0002... Ai d z A A d 92 A • • • A d2Pq.

Hence, if co is a (0, 1)-form and

co = E wa d2Œ, then

d" co =AE(Va coo — Vii coa) d? A dzs,

6" d" co= — E v. (vet coo — v„ wŒ) d2 0

5"w= — E v. coa,

d" eV co= — E vi, sic, wa ay. Hence,

LI" co = (5" d" + d" S") co

= E ( — VŒ Va (Op ± Vet Vo (pa — Vo yi of) d-ifl

=(—Va Vacopcig-FR Œpcoad20).

It is known that, for a Kahler manifold,

A = 2 z1" .

It is therefore possible to derive the fomula above for 4"w from the formula for d co.

4. A Remark on d' d"-Cohomology 159

4. A Remark on d' d" -Cohomology

We prove the following well known

Theorem 1. Let 0 and 0' be closed real (p, p)-forms on a compact Kahler manfiold M. Then 0 0' ( cohomologous to each other) if and only if there exists a real (p —1, p —1)-form 9 on M such that

0 — 0' = i d' d" (p .

Proof If 0 0' =i d' d" 9, then 0 — 0' = d (i d" (p) and hence 0 — 0'. (This implication is purely local and is valid for any complex manifold.)

To prove the converse, let ri = 0— 0' and assume ri —0 so that ti =da, where a is a real (2p 1)-form. Let a =i3- + f3, where f3 is a (p — 1, p)-form and /1 is its complex conjugate. Then

where d' )4" is of bidegree (p +1, p-1), (d" + d' 13) of bidegree (p, p), and d" fi of bidegree (p —1, p + 1). Hence,

t1=doc=d" 13, d' )6=0, d" )6=0.

We may write /3 =1-113 +d"y,

where y is a (p —1, p — 1)-form. Then

)4=11fl+d). Hence,

n=d"i3- ±d'13=d"d'7+d'd"y=d'd"(y—TI)

i di d" 9,

where (p = — i(fi — /3-) is a real (p —1, p — 1)-form.

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Index

adjoint action 140 admissible coordinate system 1, 37 affine structure 35 affine transformation 122 — —, infinitesimal 42 almost complex structure 7 almost Hamiltonian structure 11 almost Hermitian structure 10 almost symplectic structure 11 ample 83 atlas 34 —, r-atlas 34 —, maximal (complete) 34 automorphism of a Cartan connection

128 — of a G-structure 2

Bergman kernel form 78 Bergman kernel function 78 Bergman metric 78

canonical form 141, 143 canonical line bundle 83 Cartan connection 127 — —, automorphism of a 128 characteristic class 67 characteristic number 67 chart 34 complete atlas 34 complete hyperbolic 81 complete vector field 46 conformal connection 136 conformal equivalence 9, 146 conformal structure 9, 142 — —, flat 36 conformal-symplectic structure 12 conformal-symplectic transformation conformal transformation 143, 148 contact form 28 contact structure 28

contact transformation 29 — —, infinitesimal 29

degree of a G-structure 37 degree of a pseudogroup 36 degree of (compact) symmetry 55

elliptic linear Lie algebra 4, 16

filtered Lie algebra 37 — — —, transitive 37 flat conformal structure 36 flat G-structure 35 flat projective structure 36 foliation 12 frame 36, 139 fundamental vector field 127, 140

1-atlas 34 —, maximal, complete 34 r-manifold 34 r-structure 34

G-structure 1, 33 —, degree of a 37 —, flat 35 —, integrable 1 —, prolongation of a 22 general type (algebraic manifold of) 87 graded Pe algebra 38 — — —, transitive 38

Hamiltonian structure 11 — —, almost 11 hyperbolic manifold 81

27 — —, complete 81

ILH-Lie group 23 infinite type (Lie algebra of) 4

181 Index

infinitesimal — affine transformation 42 — automorphism of a G-structure — contact transformation 29 — isometry 42 — symplectic transformation 11 integrable G-structure 1, 37 intrinsic pseudo-distance 81 isometry 39 —, infinitesimal 42

Killing vector field 42

Lie pseudogroup 36

projective equivalence 145 projective structure 142

2 projective transformation 143 prolongation of a G-structure 22 — of a linear Lie algebra 4 — of a linear Lie group 19, 20 pseudogroup of transformations 34 — — —, degree of a 36 — — —, Lie 36 — —, transitive 34

residue 69 Riemann-Hurwitz relation 88

maximal atlas 34 model space 34 Möbius space 133

negative first Chern class 82 nonpositive first Chem class 103

order of a linear Lie algebra 4

parallelisable manifold 13 projective connection 136

symplectic structure 11, 23 — —, almost 11 symplectic transformation 11, 25 — —, infinitesimal 11

transitive filtered Lie algebra 37 — graded Lie algebra 38 — pseudogroup 34

very ample 78 volume element 6, 23

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